Quantum Mechanics by Konstantin K. Likharev - HTML preview
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vector of the type (37), singles out just one of its components, for example,
ˆ u u u ,
(4.43)
j
j
j
j
j
i.e. “kills” all components of the linear superposition but one. In the geometric analogy, such an operator
“projects” the state vector on the j th “direction”, hence its name – the projection operator. Probably, the most important property of the projection operators, called the closure (or “completeness”) relation, immediately follows from Eq. (41): their sum over the full basis is equivalent to the identity operator
u u
I ˆ
.
(4.44) Closure
j
j
relation
j
This means in particular that we may insert the left-hand side of Eq. (44), for any basis, into any bra-ket relation, at any place – the trick that we will use over and over again.
Now let us see how the expansions (37) transform the key notions introduced in the last section, starting with the short bracket (11), i.e. the inner product of two state vectors:
*
*
*
u u .
(4.45)
j
j
j'
j'
j
j'
jj'
j
j
j, j'
j, j'
j
Besides the complex conjugation, this expression is similar to the scalar product of the usual, geometric vectors. Now, let us explore the long bracket (23):
A ˆ * u A ˆ u * A .
(4.46)
j
j
j'
j'
j
jj'
j'
j, j'
j, j'
Here, the last form uses the very important notion of the operator’s matrix elements defined as Chapter 4
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Operator’s
matrix
A
u A ˆ
u
.
(4.47)
jj'
j
j'
elements
As Eq. (46) shows, the full set of the matrix elements completely characterizes the operator, just as the full set of the expansion coefficients (40) fully characterizes a quantum state. The term “matrix” means, first of all, that it is convenient to represent the full set of Ajj’ as a square table ( matrix), with the linear dimension equal to the number of basis states uj of the system under the consideration. By the way, this number (which may be infinite) is called the dimensionality of its Hilbert space.
As two simplest examples, all matrix elements of the null operator, defined by Eqs. (35), are evidently equal to zero (in any basis), and hence it may be represented as a matrix of zeros (called the null matrix):
0
0
Null
0 0
0 ,
(4.48)
matrix
while for the identity operator I ˆ defined by Eqs. (36), we readily get ˆ
I u I u
u u
,
(4.49)
jj'
j
j'
j
j'
jj'
i.e. its matrix (naturally called the identity matrix) is diagonal – also in any basis:
1 0 ...
Identity
I
matrix
0 1 ... .
(4.50)
... ... ...
The convenience of the matrix language extends well beyond the representation of particular operators. For example, let us use the definition (47) to calculate the matrix elements of a product of two operators:
ˆ ˆ
( AB)
u
B
A u
.
(4.51)
jj"
j
j"
Here we may use Eq. (44) for the first (but not the last!) time, inserting the identity operator between the two operators, and then expressing it via the sum of projection operators:
Matrix
element
of an
( AB)
ˆ ˆ
u
B
A u
ˆ ˆˆ
u
I
A B u
ˆ
ˆ
u A u
u B u
A B
.
(4.52)
jj"
j
j"
j
j"
operator
j
j'
j'
j"
jj' j'j"
j'
j'
product
This result corresponds to the standard “row by column” rule of calculation of an arbitrary element of the matrix product
A
A
... B
B
11
12
...
11
12
AB A
A
... B
B
.
(4.53)
21
22
...
21
22
...
...
...
...
...
...
Hence a product of operators may be represented (in a fixed basis!) by that of their matrices (in the same basis).
This is so convenient that the same language is often used to represent not only long brackets, Chapter 4
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A A ...
11
12
1
Long
ˆ
*
A A *
, *
,... A A
,
(4.54) bracket
j
jj '
j'
1
2
...
21
22
2
as a matrix
j'
... ... ...
...
product
but even short brackets:
1
Short
*
*
, *
,... ,
(4.55) bracket
j
j
1
2
2
as a matrix
j
...
product
although these equalities require the use of non-square matrices: rows of (complex-conjugate!) expansion coefficients for the representation of bra-vectors, and columns of these coefficients for the representation of ket-vectors. With that, the mapping of quantum states and operators onto matrices becomes completely general.
Now let us have a look at the outer product operator (26). Its matrix elements are just
*
u u
.
(4.56)
j
j'
j
j'
jj'
These are the elements of a very special square matrix, whose filling requires the knowledge of just 2 N
scalars (where N is the basis size) rather than N 2 scalars as for an arbitrary operator. However, a simple generalization of such an outer product may represent an arbitrary operator. Indeed, let us insert two identity operators (44), with different summation indices, on both sides of an arbitrary operator:
ˆ A ˆˆ ˆ
I I
A
ˆ
u u
A
u
u
,
(4.57)
j
j
j'
j'
j
j'
and then use the associative axiom to rewrite this expression as
A ˆ u u A ˆ u
u .
(4.58)
j
j
j'
j'
j, j'
But the expression in the middle long bracket is just the matrix element (47), so we may write Operator
ˆ A u A u .
(4.59) via its
j
jj '
j'
matrix
j, j'
elements
The reader should agree that this formula, which is a natural generalization of Eq. (44), is extremely elegant.
The matrix representation is so convenient that it makes sense to extend it to one level lower –
from the state vector products to the “bare” state vectors resulting from the operator’s action upon a given state. For example, let us use Eq. (59) to represent the ket-vector (18) as
' ˆ A u A u
u A u .
(4.60)
j
jj '
j'
j jj' j'
j, j'
j, j'
According to Eq. (40), the last short bracket is just j’, so
' u A
(4.61)
'
'
A
u
j
jj
j
jj'
j'
j
j, j'
j j'
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But the expression in the parentheses is just the coefficient ’j of the expansion (37) of the resulting ket-vector (60) in the same basis, so
'
A .
(4.62)
j
jj' j'
j'
This result corresponds to the usual rule of multiplication of a matrix by a column, so we may represent any ket-vector by its column matrix, with the operator’s action looking like
'
A
A
1
...
11
12
1
'
A
A
.
(4.63)
2
...
21
22
2
... ...
...
...
...
Absolutely similarly, the operator action on the bra-vector (21), represented by its row matrix, is
†
†
A
A
...
11
12
' ,
*
*
*
*
†
†
' ,
A
A
.
(4.64)
1
2
... ,1 2
,... ...
21
22
...
...
...
By the way, Eq. (64) naturally raises the following question: what are the elements of the matrix on its right-hand side, or more exactly, what is the relation between the matrix elements of an operator and its Hermitian conjugate? The simplest way to answer it is to use Eq. (25) with two arbitrary states (say, uj and uj’) of the same basis in the role of and . Together with the orthonormality relation (38), this immediately gives15
Hermitian
conjugate:
†
ˆ
*
A A
.
(4.65)
jj'
j' j
matrix
elements
Thus, the matrix of the Hermitian-conjugate operator is the complex conjugated and transposed matrix of the initial operator. This result exposes very clearly the difference between Hermitian and complex conjugation. It also shows that for the Hermitian operators defined by Eq. (22),
*
A A ,
(4.66)
jj'
j' j
i.e. any pair of their matrix elements, symmetric with respect to the main diagonal, should be the complex conjugate of each other. As a corollary, their main-diagonal elements have to be real:
*
A A ,
Im
i.e.
A .
0
(4.67)
jj
jj
jj
15 For the sake of formula compactness, below I will use the shorthand notation in that the operands of this equality are just A† jj’ and A* j’j. I believe that it leaves little chance for confusion, because the Hermitian conjugation sign † may pertain only to an operator (or its matrix), while the complex conjugation sign *, to a scalar – say a matrix element.
Chapter 4
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In order to fully appreciate the special role played by Hermitian operators in quantum theory, let us introduce the key notions of eigenstates aj (described by their eigenvectors aj and aj) and eigenvalues ( c-numbers) Aj of an operator A ˆ , both defined by the equation they have to satisfy:16
Operator:
A ˆ a A a .
(4.68) eigenstates
j
j
j
and
eigenvalues
Let us prove that the eigenvalues of any Hermitian operator are real,17
Hermitian
*
A A , for j ,
1 ,...,
2
N,
(4.69) operator:
j
j
eigenvalues
while the eigenstates corresponding to different eigenvalues are orthogonal:
Hermitian
a a
,
0
if A A .
(4.70)
j
j'
j
j'
operator:
eigenvectors
The proof of both statements is surprisingly simple. Let us inner-multiply both sides of Eq. (68) by the bra-vector aj’. On the right-hand side of the result, the eigenvalue Aj, as a c-number, may be taken out of the bracket, giving
a A ˆ a A a a .
(4.71)
j'
j
j
j'
j
This equality has to hold for any pair of eigenstates, so we may swap the indices j and j’ in Eq. (71), and write the complex-conjugate of the result:
*
ˆ
*
*
a A a
A a a
.
(4.72)
j
j'
j'
j
j'
Now using Eqs. (14) and (25), together with the Hermitian operator’s definition (22), we may transform Eq. (72) into the following form:
a A ˆ a A* a a .
(4.73)
j'
j
j'
j'
j
Subtracting this equation from Eq. (71), we get
0
*
A A
a a .
(4.74)
j
j'
j'
j
There are two possibilities to satisfy this relation. If the indices j and j’ are equal (denote the same eigenstate), then the bracket is the state’s norm squared, and cannot be equal to zero. In this case, the left parentheses (with j = j’) have to be zero, proving Eq. (69). On the other hand, if j and j’
correspond to different eigenvalues of A, the parentheses cannot equal zero (we have just proved that all Aj are real!), and hence the state vectors indexed by j and j’ should be orthogonal, e.g., Eq. (70) is valid.
As will be discussed below, these properties make Hermitian operators suitable, in particular, for the description of physical observables.
16 This equation should look familiar to the reader – see the stationary Schrödinger equation (1.60), which was the focus of our studies in the first three chapters. We will see soon that that equation is just a particular (coordinate) representation of Eq. (68) for the Hamiltonian as the operator of energy.
17 The reciprocal statement is also true: if all eigenvalues of an operator are real, it is Hermitian (in any basis).
This statement may be readily proved by applying Eq. (93) below to the case when Akk’ = Ak kk’, with Ak* = Ak.
Chapter 4
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4.4. Change of basis, and matrix diagonalization
From the discussion of the last section, it may look like the matrix language is fully similar to, and in many instances more convenient than the general bra-ket formalism. In particular, Eqs. (54)-(55) and (63)-(64) show that any part of any bra-ket expression may be directly mapped onto the similar matrix expression, with the only slight inconvenience of using not only columns but also rows (with their elements complex-conjugated), for state vector representation. This invites the question: why do we need the bra-ket language at all? The answer is that the matrix elements depend on the particular choice of the basis set, very much like the Cartesian components of a usual geometric vector depend on the particular choice of reference frame orientation (Fig. 4), and very frequently, at problem solution, it is convenient to use two or more different basis sets for the same system. (Just a bit more patience –
numerous examples will follow soon.)
y
y'
α
y
'y
Fig. 4.4. The transformation
x'
'
of components of a 2D vector
x
at a reference frame’s rotation.
0
x
x
With this motivation, let us explore what happens at the transform from one basis, { u}, to another one, { v} – both full and orthonormal. First of all, let us prove that for each such pair of bases, and an arbitrary numbering of the states of each base, there exists such an operator U ˆ that, first, v
U ˆ
u ,
(4.75)
j
j
Unitary
operator: and, second,
definition
ˆ U
U ˆ † U ˆ U
† ˆ I ˆ.
(4.76)
(Due to the last property,18 U îs called a unitary operator, and Eq. (75), a unitary transformation.) A very simple proof of both statements may be achieved by construction. Indeed, let us take Unitary
operator:
U ˆ
construction
v u ,
(4.77)
j'
j'
j'
- an evident generalization of Eq. (44). Then, using Eq. (38), we obtain
U ˆ u
v u u v v ,
(4.78)
j
j'
j'
j
j'
j'j
j
j'
j'
so Eq. (75) has been proved. Now, applying Eq. (31) to each term of the sum (77), we get Unitary
operator:
U †
ˆ
conjugate
u v ,
(4.79)
j'
j'
j'
18 An alternative way to express Eq. (76) is to write
1
ˆ †
ˆ
U U , but I will avoid using this language.
Chapter 4
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so
U
U †
ˆ ˆ v u u v
.
(4.80)
'
v
v
'
v v
j
j
j'
j
j
jj'
j
j
j
j, j'
j, j'
j
But according to the closure relation (44), the last expression is just the identity operator, so one of Eqs.
(76) has been proved. (The proof of the second equality is absolutely similar.) As a by-product of our proof, we have also got another important expression – Eq. (79). It implies, in particular, that while, according to Eq. (75), the operator U ˆ performs the transform from the “old” basis { u} to the “new”
basis { v}, its Hermitian adjoint †
ˆ
U performs the reciprocal transform:
Reciprocal
ˆ †
U v u u .
(4.81) basis
j
j'
j'j
j
transform
j'
Now let us see what the matrix elements of the unitary transform operators look like. Generally, as was discussed above, the operator’s elements may depend on the basis we calculate them in, so let us be specific – at least initially. For example, let us calculate the desired matrix elements Ujj’ in the “old”
basis { u}, by using Eq. (77):
ˆ
U
u U u
u
v u
u
u v u v . (4.82)
jj' in u
j
j'
j
j"
j"
j'
j
j"
j"j'
j
j'
j"
j"
Now performing a similar calculation in the “new” basis { v}, we get
ˆ
U
v U v
v
v u
v
u v u v .
(4.83)
jj' in v
j
j'
j
j"
j"
j'
jj"
j"
j'
j
j'
j"
j"
Surprisingly, the result is the same! This is of course true for the Hermitian conjugate (79) as well:
†
†
U
U
v u .
(4.84)
jj ' in u
jj ' in v
j
j'
These expressions may be used, first of all, to rewrite Eq. (75) in a purely matrix form. Applying the first of Eqs. (41) to any state vj’ of the “new” basis, and then Eq. (82), we get v
u
u v
U u
.
(4.85)
j'
j
j
j'
jj' j
j
j
Basis
transforms:
Similarly, the reciprocal transform is
matrix
form
u
v
v u
U † v
.
(4.86)
j'
j
j
j'
jj' j
j
j
These formulas are very convenient for applications; we will use them already in this section.
Next, we may use Eqs. (83)-(84) to express the effect of the unitary transform on the expansion coefficients j of the vectors of an arbitrary state , defined by Eq. (37). As a reminder, in the “old”
basis { u} they are given by Eqs. (40). Similarly, in the “new” basis { v},
v .
(4.87)
j in v
j
Again inserting the identity operator in its closure form (44) with the internal index j’, and then using Eqs. (84) and (40), we get
Chapter 4
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v u u
v u u U † u U†
. (4.88)
j
v
in
j
j'
j'
j
j'
j'
jj'
j'
jj'
j'
u
in
j'
j'
j'
j'
The reciprocal transform is performed by matrix elements of the operator U ˆ :
U
.
(4.89)
j
u
in
jj'
j'
v
in
j'
Per Eqs. (82)-(84), the matrix elements U and †
U
are the same in the bases { u} and { v}, so Eqs.
jj '
jj'
(88)-(89) may be rewritten in the following compact matrix form:
U†
,
U
,
(4.90)
in v
in u
in u
in v
even though the reader should remember that these relations are different from the usual matrix formulas, which use the same basis for all its components.
So, if the transform (75) from the “old” basis { u} to the “new” basis { v} is performed by a unitary operator, the change (88) of state vector components at this transformation requires its Hermitian conjugate. This fact is similar to the transformation of components of a usual vector at coordinate frame rotation. For example, for a 2D vector whose actual position in space is fixed (Fig. 4):
'
x
cos
sin
x ,
(4.91)
'
y
sin
cos
y
but the reciprocal transform is performed by a different matrix, which may be obtained from that participating in Eq. (91) by the replacement –. This replacement has a clear geometric sense: if the “new” reference frame { x’, y’} is obtained from the “old” frame { x, y} by a counterclockwise rotation by angle , the reciprocal transformation requires such rotation with angle –. (In this analogy, the unitary property (76) of the unitary transform operators corresponds to the equality of the determinants of both rotation matrices to 1.)
Now let us use the same trick of identity operator insertion, repeated twice, to find the transformation rule for matrix elements of an arbitrary operator:
A
v A ˆ v v
u
u
A
u
u
v
U †
ˆ
A
U
;
(4.92)
jj' in v
j
j'
j
k
k
k'
k'
j'
in
k
k'
jk
kk'
u
k'j'
k , k'
Matrix
elements’ absolutely similarly, we may also get
transforms
A
U A
U † .
(4.93)
jj' in u
jk kk' in v k'j'
k , k'
In the spirit of Eq. (90), we may represent these results in the similar matrix form:
†
†
A
U A
U,
A
UA
U ,
(4.94)
in v
in u
in u
in v
where, again, the matrix elements of
†
U
and
U
may be calculated in any of the bases { u} and { v} – but
not in an arbitrary basis!
As a sanity check, let us apply Eq. (93) to the identity operator:
Chapter 4
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I ˆ
U ˆ † IU
ˆ ˆ
U ˆ U
† ˆ
I ˆ
(4.95)
v
in
in u
u
in
in u
– just as it should be. One more invariant of the basis change is the trace of any operator, defined as the sum of the diagonal terms of its matrix:
A ˆ
Tr
A
Tr A .
(4.96) Operator/
jj
matrix:
j
trace
The (easy) proof of this fact, using previous relations, is left for the reader’s exercise.
So far, I have implied that both state bases { u} and { v} are known, and the natural question is where this information comes from in the quantum mechanics of actual physical systems. To get a partial answer to this question, let us return to Eq. (68), which defines the eigenstates and the eigenvalues of an operator. Let us assume that the eigenstates aj of a certain operator A ˆ form a full and orthonormal set, and calculate the matrix elements of the operator in the basis { a} of these states, at their arbitrary numbering. For that, it is sufficient to inner-multiply both sides of Eq. (68), written for some eigenstate aj’, by the bra-vector of an arbitrary state aj of the same set: a A ˆ a
a A a .
(4.97)
j
j'
j
j'
j'
The left-hand side of this equality is the matrix element Ajj’ we are looking for, while its right-hand side is just Aj’ jj’. As a result, we see that the matrix is diagonal, with the diagonal consisting of the operator’s eigenvalues:
Matrix
A A .
(4.98) elements in
jj'
j
jj'
eigenstate
basis
In particular, in the eigenstate basis (but not necessarily in an arbitrary basis!), A jj means the same as Aj.
Thus the important problem of finding the eigenvalues and eigenstates of an operator is equivalent to the diagonalization of its matrix,19 i.e. finding the basis in which the operator’s matrix acquires the diagonal form (98); then the diagonal elements are the eigenvalues, and the basis itself is the desirable set of eigenstates.
To see how this is done in practice, let us inner-multiply Eq. (68) by a bra-vector of the basis (say, { u}) in that we have happened to know the matrix elements Ajj’: u A ˆ a u A a .
(4.99)
k
j
k
j
j
On the left-hand side, we can (as usual :-) insert the identity operator between the operator A ând the ket-vector, and then use the closure relation (44) in the same basis { u}, while on the right-hand side, we can move the eigenvalue Aj (a c-number) out of the bracket, and then insert a summation over the same index as in the closure, compensating it with the proper Kronecker delta symbol: u A ˆ u u a A u a .
(4.100)
k
k'
k'
j
j
k '
j
kk'
k'
k '
Moving out the signs of summation over k’, and using the definition (47) of the matrix elements, we get 19 Note that the expression “matrix diagonalization” is a very common but dangerous jargon. Formally, a matrix is just a table, an ordered set of c-numbers, and cannot be “diagonalized”. It is OK to use this jargon (I will do this) if you remember clearly what it actually means – see the definition above.
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A A
u a .
(4.101)
kk '
j
kk '
0
k '
j
k '
But the set of such equalities, for all N possible values of the index k, is just a system of homogeneous linear equations for unknown c-numbers uk’ aj. According to Eqs. (82)-(84), these numbers are nothing else than the matrix elements Uk’j of a unitary matrix providing the required transformation from the initial basis { u} to the basis { a} that diagonalizes the matrix A. This system may be represented in the matrix form:
Matrix
A A
A
... U
j
j
11
12
1
diagonali-
zation
A
A A
... U
,
(4.102)
j
j 0
21
22
2
...
...
... ...
and the condition of its consistency,
A A
A
...
Characteristic
11
j
12
equation
A
A A
... ,
0
(4.103)
for
21
22
j
eigenvalues
...
...
...
plays the role of the characteristic equation of the system. This equation has N roots Aj – the eigenvalues of the operator A ˆ ; after they have been calculated, plugging any of them back into the system (102), we can use it to find N matrix elements Ukj ( k = 1, 2, … N) corresponding to this particular eigenvalue.
However, since the equations (102) are homogeneous, they allow finding Ukj only to a constant multiplier. To ensure their normalization, i.e. enforce the unitary character of the matrix U, we may use the requirement for all eigenvectors to be normalized (just as the basis vectors are): 2
a a
a u
u a
U
(4.104)
j
j
j
k
k
j
,
1
kj
k
k
for each j. This normalization completes the diagonalization.20
Now (at last!) I can give the reader some examples. As a simple but very important case, let us diagonalize each of the operators described (in a certain two-function basis { u}, i.e. in two-dimensional Hilbert space) by the so-called Pauli matrices
0 1
0 i
1 0
Pauli
σ
,
σ
,
σ
.
(4.105)
x
matrices
1 0
y
i
0
z
0 1
Though introduced by a physicist, with a specific purpose to describe the electron’s spin, these matrices have a general mathematical significance, because together with the 22 identity matrix, they provide a full, linearly-independent system – meaning that an arbitrary 22 matrix may be represented as
A
A
11
12
b I c σ c σ c σ ,
(4.106)
x
x
y
y
z
z
A
A
21
22
20 A possible slight complication here is that the characteristic equation may give equal eigenvalues for certain groups of different eigenvectors. In such cases, the requirement of the mutual orthogonality of these degenerate states should be additionally enforced.
Chapter 4
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with a unique set of four c-number coefficients b, cx, cy, and cz.
Since the matrix z is already diagonal, with the evident eigenvalues 1, let us start with diagonalizing the matrix x. For it, the characteristic equation (103) is evidently
A
1
j
,
0
i.e.
2
A 1 ,
0
(4.107)
1
j
Aj
and has two roots, A 1,2 = ±1. (Again, the state numbering is arbitrary!) So the eigenvalues of the matrix
x are the same as those of the matrix z. (The reader may readily check that the eigenvalues of the matrix y are also the same.) However, the eigenvectors of the operators corresponding to these three matrices are different. To find them for x, let us plug its first eigenvalue, A 1 = +1, back into equations (101) spelled out for this particular case ( j = 1; k, k’ = 1,2):
u a u a ,
0
1
1
2
1
(4.108)
u a u a .
0
1
1
2
1
These two equations are compatible (of course, because the used eigenvalue A 1 = +1 satisfies the characteristic equation), and any of them gives
u a u a ,
e.
i.
U U .
(4.109)
1
1
2
1
11
21
With that, the normalization condition (104) yields
2
2
1
U
U
.
(4.110)
11
21
2
Although the normalization is insensitive to the simultaneous multiplication of U 11 and U 21 by the same phase factor exp{ i} with any real , it is convenient to keep the coefficients real, for example taking
= 0, to get
1
U U
.
(4.111)
11
21
2
Performing an absolutely similar calculation for the second characteristic value, A 2 = –1, we get U 12 = – U 22, and we may choose the common phase to have
1
U U
,
(4.112)
12
22
2
so the whole unitary matrix for diagonalization of the operator corresponding to x is21
†
1 1
1
Unitary matrix
U U
,
(4.113)
x
x
diagonalizing
2 1 1
x
For what follows, it will be convenient to have this result expressed in the ket-relation form – see Eqs.
(85)-(86):
1
1
a U u U u
u u
a U u U u
u u
(4.114a)
1
11
1
21
2
1
2 ,
2
12
1
22
2
1
2 ,
2
2
21 Though this particular unitary matrix U x is Hermitian, this is not true for an arbitrary choice of the phases .
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†
†
1
†
†
1
u U a U a
a a
u U a U
a
a a
(4.114b)
1
11
1
21
2
1
2 ,
2
12
1
22
2
1
2 .
2
2
Now let me show that these results are already sufficient to understand the Stern-Gerlach experiments described in Sec. 1 – but with two additional postulates. The first of them is that the interaction of a particle with the external magnetic field, besides that due to its orbital motion, may be described by the following operator vector of its spin dipole magnetic moment:22
Spin
magnetic
m
ˆ Sˆ
,
(4.115a)
moment
where the constant coefficient , specific for every particle type, is called the gyromagnetic ratio,23 and Sîs the operator vector 24 of spin, with three Cartesian components: Spin
vector
ˆS n S ˆ n S ˆ n S ˆ .
(4.115b)
x
x
y
y
z
z
operator
Here n x,y,z are the usual Cartesian unit vectors in the 3D geometric space (in the quantum-mechanics sense, they are just c-numbers, or rather “c-vectors”), while S âre the “usual” (scalar) operators. For x, y, z
the so-called spin-½ particles (including the electron),25 these components may be simply, as S ˆ
σ ˆ
,
(4.116a)
x, y, z
x, y, z
2
Pauli expressed via those of the Pauli vector ˆσ n ˆ n ˆ n ˆ , so we may also write x
x
y
y
z
z
vector
Sˆ
σˆ .
(4.116b)
2
In turn, in the so-called z-basis, each Cartesian component of the latter operator is just the corresponding Pauli matrix (105), so it may be also convenient to use the following 3D vector of these matrices:26
Pauli
n
n in
z
x
y
vector’s
σ n σ n σ n σ
.
(4.117)
x
x
y
y
z
z
matrix
n in
n
x
y
z
The z-basis, in which such matrix representation of ˆ
σ is valid, is defined as an orthonormal basis
of certain two states, commonly denoted (“spin up”) an (“spin down”). In this basis, the matrix of the operator σ îs diagonal, with eigenvalues, respectively, + 1 and –1, and hence the matrix S
z
z
(/2) z of S îs also diagonal with the eigenvalues +
z
/2 and –/2 – see the last of Eqs. (105). Note that
22 This was the key point in the electron spin’s description, developed by W. Pauli in 1925-1927.
23 For the electron, with its negative charge q = – e, the gyromagnetic ratio is negative: e = – g e e/2 m e, where g e
2 is the electron’s dimensionless g-factor. Due to quantum-electrodynamic (relativistic) effects, this g-factor is slightly higher than 2: g e = 2(1 + /2 + …) 2.002319304…, where e 2/40 c ( E H/ m e c 2)1/2 1/137 is the so-called fine structure constant. (The origin of its name will be clear from the discussion in Sec. 6.3.) 24 The basic rule of dealing with operator vectors is to perform all vector operations just as with the usual geometric vectors. (The vector is a good example – see the formulas for in MA Secs. 8-12.) 25 At this point, the adjective “spin-½ ” should be understood as just a name. The physical sense of this term and the generalization of the theory to other values of spin will be discussed in Sec. 5.7.
26 Note that is some texts, the term “Pauli vector” is used for this matrix rather than for the operator σˆ .
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we do not “understand” what exactly the states and are,27 but loosely associate them with some internal rotation of a spin-½ particle about the z-axis, with either positive or negative angular momentum component Sz. However, attempts to use such classical interpretation for quantitative predictions run into fundamental difficulties – see Sec. 6 below.
The second necessary postulate describes the general relation between the bra-ket formalism and experiment. Namely, in quantum mechanics, each real observable A is represented by a Hermitian operator ˆ
ˆ †
A A , and the result of its measurement,28 in a quantum state described by a linear superposition of the eigenstates aj of the operator,
a , with a ,
(4.118)
j
j
j
j
j
may be only one of the corresponding eigenvalues Aj.29 Specifically, if the ket (118) and all eigenkets
aj are normalized to 1,
,
1
a a
1,
(4.119)
j
j
then the probability of a certain measurement outcome Aj is30
2
Quantum
W
*
a a ,
(4.120) measurement
j
j
j
j
j
j
postulate
This relation is evidently a generalization of Eq. (1.22) in wave mechanics. As a sanity check, let us assume that the set of the eigenstates aj is full, and calculate the sum of the probabilities to find the system in each of these states:
ˆ
W
.
(4.121)
j
a a I 1
j
j
j
j
Now returning to the Stern-Gerlach experiment, conceptually the description of the first ( z-
oriented) experiment shown in Fig. 1 is formally the hardest for us, because the statistical ensemble describing the unpolarized particle beam at its input is mixed (“incoherent”), and cannot be described by a pure (“coherent”) superposition of the type (6) that have been the subject of our studies so far. (We will discuss mixed ensembles in Chapter 7.) However, it is intuitively clear that its results are compatible with the description of the two output beams as sets of particles in the pure states and , respectively. The absorber following that first stage (Fig. 2) just takes all spin-down particles out of the picture, producing an output beam of polarized particles in the definite state. For such a beam, the 27 If you think about it, the word “understand” typically means that we can express a new notion in terms of those discussed earlier and thus considered “known”. (For example, in our current case, we cannot describe the spin states by any wavefunction (r), or any other mathematical notion discussed in the previous three chapters and hence considered “known”.) The bra-ket formalism was invented exactly to enable mathematical analyses of such
“new” quantum states we do not initially “understand”. Gradually, as we learn more and more about their properties and get accustomed to these notions, we start treating them as “known” ones.
28 Here again, just like in Sec. 1.2, the statement implies the abstract notion of “ideal experiments”, deferring the discussion of real (physical) measurements until Chapter 10.
29 As a reminder, at the end of Sec. 3 we have already proved that such eigenstates corresponding to different values Aj are orthogonal. If any of these values is degenerate, i.e. corresponds to several different eigenstates, they should be also selected orthogonal, in order for Eq. (118) to be valid.
30 This relation, in particular, explains the most common term for the (generally, complex) coefficients j, which was already mentioned several times earlier: the probability amplitudes.
Chapter 4
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probabilities (120) are W = 1 and W = 0. This is certainly compatible with the result of the “control”
experiment shown on the bottom panel of Fig. 2: the repeated SG ( z) stage does not split such a beam, keeping the probabilities the same.
Now let us discuss the double Stern-Gerlach experiment shown on the top panel of Fig. 2. For that, let us represent the z-polarized beam in another basis – of the two states (I will denote them as
and ) in that, by definition, the matrix S x is diagonal. But this is exactly the set we called a 1,2 in the x matrix diagonalization problem solved above. On the other hand, the states and are exactly what we called u 1,2 in that problem because in this basis, we know the matrix explicitly – see Eq. (117). Hence, in the application to the particle spin problem, we may rewrite Eqs. (114) as
1
1
Relation
,
,
(4.122)
between
2
2
eigenvectors
of Sx and Sz
1
1
,
,
(4.123)
2
2
Currently for us the first of Eqs. (123) is most important, because it shows that the quantum state of particles entering the SG ( x) stage may be represented as a coherent superposition of particles with Sx
= +/2 and Sx = –/2. Notice that the beams have equal probability amplitude moduli, so according to Eq. (120), the split beams and have equal intensities, in accordance with experimental results.
Now, let us discuss the most mysterious (from the classical point of view) multistage SG
experiment shown on the middle panel of Fig. 2. After the second absorber has taken out all particles in, say, the state, the remaining particles, all in the state , are passed to the final, SG ( z), stage. But according to the first of Eqs. (122), this state may be represented as a (coherent) linear superposition of the and states, with equal probability amplitudes. The final stage separates particles in these two states into separate beams, with equal probabilities W = W = ½ to find an particle in each of them, thus explaining the experimental results.
To conclude our discussion of the multistage Stern-Gerlach experiment, let me note that though it cannot be explained in terms of wave mechanics (which operates with scalar de Broglie waves), it has an analogy in classical theories of vector fields, such as the classical electrodynamics. Indeed, let a plane electromagnetic wave propagate normally to the plane of the drawing in Fig. 5, and pass through the linear polarizer 1.
1
/4
2
Fig. 4.5. A light polarization sequence similar to the three-stage
3
Stern-Gerlach experiment shown on the middle panel of Fig. 2.
0
Similarly to the output of the initial SG ( z) stages (including the absorbers) shown in Fig. 2, the output wave is linearly polarized in one direction – the vertical direction in Fig. 5. Now its electric field vector has no horizontal component – as may be revealed by the wave’s full absorption in a Chapter 4
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perpendicular polarizer 3. However, let us pass the wave through polarizer 2 first. In this case, the output wave does acquire a horizontal component, as can be, again, revealed by passing it through polarizer 3. If the angles between the polarization directions 1 and 2, and between 2 and 3, are both equal to /4, each polarizer reduces the wave amplitude by a factor of 2, and hence the intensity by a factor of 2, exactly like in the multistage SG experiment, with the polarizer 2 playing the role of the SG
( x) stage. The “only” difference is that the necessary angle between the polarizer orientations is /4, rather than /2 for the Stern-Gerlach experiment. In quantum electrodynamics (see Chapter 9 below), which confirms classical predictions for this experiment, this difference may be explained by that between the integer spin of electromagnetic field quanta (photons) and the half-integer spin of electrons.
4.5. Observables: Expectation values and uncertainties
After this particular (and hopefully inspiring) example, let us discuss the general relation between the Dirac formalism and experiment in more detail. The expectation value of an observable over any statistical ensemble (not necessarily a coherent one) may be always calculated using the general statistical rule (1.37). For the particular case of a coherent superposition (118), we can combine that rule with Eq. (120) and the second of Eqs. (118):
A
A W
* A
a A a
a A a
.
(4.124)
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
Now using Eq. (59) for the particular case of the eigenstate basis { a}, for which Eq. (98) is valid, we arrive at a very simple and important formula31
Expectation
A
A ˆ
.
(4.125) value
as a long
bracket
This is a clear analog of the wave-mechanics formula (1.23) – and as we will see soon, may be used to derive it.32 A great convenience of Eq. (125) is that it does not explicitly involve the eigenvector set of the corresponding operator, and allows the calculation to be performed in any convenient basis.
For example, let us consider an arbitrary coherent state of spin-½,33 and calculate the expectation values of its components. The calculations are easier in the z-basis because we know the matrix elements of the spin operator components in that basis. Representing the ket- and bra-vectors of the given state as linear superpositions of the corresponding vectors of the basis states and ,
*
*
,
.
(4.126)
and plugging these expressions into Eq. (125) written for the observable Sz, we get 31 This equality reveals the full beauty of Dirac’s notation. Indeed, initially in this chapter, the quantum-mechanical brackets just reminded the angular brackets used for statistical averaging. Now we see that in this particular (but most important) case, the angular brackets of these two types may be indeed equal to each other!
32 Note also that Eq. (120) may be rewritten in a form similar to Eq. (125): W ˆ , where îs the j
j
j
operator (42) of the state’s projection upon the j th eigenstate aj.
33 For clarity, the noun “spin-½” is used, here and below, to denote the spin degree of freedom of a spin-½
particle, independent of its orbital motion.
Chapter 4
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*
* ˆ
S
S
z
z
(4.127)
*
ˆ
*
ˆ
*
ˆ
*
ˆ
S S S S .
z
z
z
z
Now there are two equivalent ways (both very simple) to calculate the long brackets in this expression. The first one is to represent each of them in the matrix form in the z-basis, in which the bra-and ket-vectors of states and are the matrix rows (1, 0) and (0, 1), or similar matrix columns – the exercise highly recommended to the reader. Another (perhaps more elegant) way is to use the general Eq. (59), in the z-basis, together with the spin-½-specific Eqs. (116a) and (105) to write Spin-½
component
ˆ S
S
i
S
.
(4.128)
x
ˆ
,
y
ˆ
,
z
operators
2
2
2
For our particular calculation, we may plug the last of these expressions into Eq. (127), and use the orthonormality conditions (38):
,
1
0 .
(4.129)
Both approaches give (of course) the same result:
S
*
*
.
(4.130)
z
2
This particular result might be also obtained using Eq. (120) for the probabilities W = *
and W = *, namely:
*
*
S W
W
.
(4.131)
z
2
2
2
2
The formal way (127), based on the general Eq. (125), has, however, the advantage of being applicable to finding the observables whose operators are not diagonal in the z-basis, as well. In particular, absolutely similar calculations give
*
ˆ
*
ˆ
*
ˆ
*
ˆ
*
*
S S S S S
(4.132)
x
x
x
x
x
,
2
*
ˆ
*
ˆ
*
ˆ
*
ˆ
*
*
S
S S S S i
(4.133)
y
y
y
y
y
,
2
Let us have a good look at a particular spin state, for example the spin-up state . According to Eq. (126), in this state = 1 and = 0, so Eqs. (130)-(133) yield:
S ,
S S
0 .
(4.134)
z
2
x
y
Now let us use the same Eq. (125) to calculate the spin component uncertainties. According to Eqs.
(105) and (116)-(117), the operator of each spin component squared is equal to (/2)2 I ˆ , so the general Eq. (1.33) yields
Chapter 4
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2
2
2
S
(4.135a)
z 2
2
2
ˆ 2
ˆ
S S
S
z
z
z
I ,
0
2
2
2
2
2
S
(4.135b)
x 2
2
2
ˆ 2
ˆ
S S
S 0
x
x
x
I ,
2
2
2
2
S
.
(4.135c)
y
2
2
2
2
ˆ
ˆ
S S
S 0
y
y
y
I
2
2
While Eqs. (134) and (135a) are compatible with the classical notion of the angular momentum of magnitude /2 being directed exactly along the z-axis, this correspondence should not be overstretched, because such a classical picture cannot explain Eqs. (135b) and (135c). The best (but still imprecise!) classical image I can offer is the spin vector S oriented, on average, in the z-direction, but still having its x- and y-components strongly “wobbling” (fluctuating) about their zero average values.
It is straightforward to verify that in the x-polarized and y-polarized states, the situation is similar, with the corresponding change of axis indices. Thus, in neither of these states, all three spin components have definite values. Let me show that this is not just an occasional fact, but reflects one of the most profound properties of quantum mechanics, the uncertainty relations. For that, let us consider two measurable observables, A and B, of the same quantum system. There are two possibilities here. If the operators corresponding to these observables commute,
ˆ ˆ
,
A B 0 ,
(4.136)
then all matrix elements of the commutator in any orthogonal basis (in particular, in the basis of eigenstates aj of the operator A ˆ ) have to equal zero:
a
A B a
a AB a
a BA a
.
(4.137)
j ˆ
ˆ
,
ˆ ˆ
ˆ ˆ
0
j '
j
j '
j
j '
In the first bracket of the middle expression, let us act by the (Hermitian!) operator A ôn the bra-vector, while in the second one, on the ket-vector. According to Eq. (68), such action turns the operators into the corresponding eigenvalues, which may be taken out of the long brackets, so we get ˆ
ˆ
ˆ
A a B a
A a B a
A
A
a B a
(4.138)
j
j
j'
j'
j
j'
j
j'
0.
j
j'
This means that if all eigenstates of the operator A âre non-degenerate (i.e. Aj Aj’ if j j’), the matrix of the operator B ˆ has to be diagonal in the basis { a}, i.e., the operators A ând B ˆ have common eigenstates. Such pairs of observables (and their operators) that can share their eigenstates are called compatible. For example, in the wave mechanics of a particle, its momentum (1.26) and kinetic energy (1.27) are compatible, sharing their eigenfunctions (1.29). Now we see that this is not occasional, because each Cartesian component of the kinetic energy is proportional to the square of the corresponding component of the momentum, and any operator commutes with an arbitrary integer power of itself:
ˆ ˆ,
A An
ˆ ˆ ˆ ˆ
ˆ ˆ ˆ ˆ
ˆ ˆ ˆ ˆ
,
A
...
A
A
A A
...
A
A
A
...
A
A
AA 0 .
(4.139)
n
n
n
Chapter 4
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Now, what if the operators A ând B ˆ do not commute? Then the following general uncertainty relation is valid:
General
1
uncertainty
A
B
A ˆ B ˆ, ,
(4.140)
relation
2
where all expectation values are for the same but arbitrary state of the system. The proof of Eq. (140) may be divided into two steps, the first one proving the so-called Schwartz inequality for any two possible states, say and :34
Schwartz
inequality
2
.
(4.141)
Its proof may be readily achieved by applying the postulate (16) – that the norm of any legitimate state of the system cannot be negative – to the state with the following ket-vector:
,
(4.142)
where and are possible, non-null states of the system, so the denominator in Eq. (142) is not equal to zero. For this case, Eq. (16) gives
.
0
(4.143)
Opening the parentheses, we get
0.
(4.144)
2
After the cancellation of one inner product in the numerator and the denominator of the last term, it cancels with the 2nd (or the 3rd) term. What remains is the Schwartz inequality (141).
Now let us apply this inequality to states
A ˆ
~
and
B ˆ~
,
(4.145)
where, in both relations, is the same possible state of the system, and the deviation operators are defined similarly to the deviations of the observables (see Sec. 1.2):
~
A ˆ~ A ˆ A
B ˆ
,
B ˆ B .
(4.146)
With this substitution, and taking into account again that the observable operators A ând B âre Hermitian, Eq. (141) yields
2
ˆ~2
ˆ~2
ˆ~ ˆ~
A B AB .
(4.147)
Since the state is arbitrary, we may use Eq. (125) to rewrite this relation as an operator inequality: 34 This inequality is the quantum-mechanical analog of the usual vector algebra’s result 22 2.
Chapter 4
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A
B
A ˆ
~
B ˆ~ .
(4.148)
Actually, this is already an uncertainty relation, even “better” (stronger) than its standard form (140); moreover, it is more convenient in some cases. To prove Eq. (140), we need a couple of more steps. First, let us notice that the operator product participating in Eq. (148) may be recast as ˆ~ ˆ~ 1 ˆ~ ˆ~
i
ˆ
ˆ
ˆ~ ˆ~
AB
A, B C,
C
where
i A, B .
(4.149)
2
2
Any anticommutator of Hermitian operators, including that in Eq. (149), is a Hermitian operator, and its eigenvalues are purely real, so its expectation value (in any state) is also purely real. On the other hand, the commutator part of Eq. (149) is just
C i
A ˆ~
ˆ
B ˆ~
, i A ˆ A B ˆ B i B ˆ B A ˆ A i ˆ B
A ˆ
ˆ
A
B ˆ i A ˆ B ˆ
,
.
(4.150)
Second, according to Eqs. (52) and (65), the Hermitian conjugate of any product of the Hermitian operators A ând B îs just the product of these operators swapped. Using this fact, we may write
†
C ˆ † i A ˆ B ˆ
, i A ˆ
( B ˆ)† i B ˆ
( A ˆ † iB ˆ
)
A ˆ iA ˆ B ˆ i A ˆ B ˆ
, C ˆ ,
(4.151)
so the operator C îs also Hermitian, i.e. its eigenvalues are also real, and thus its expectation value is purely real as well. As a result, the square of the expectation value of the operator product (149) may be represented as
2
2
2
ˆ~ ˆ~
1 ˆ~ ˆ~
1 ˆ
AB
A, B
C
2
.
(4.152)
2
Since the first term on the right-hand side of this equality cannot be negative, we may write 2
2
ˆ~ ˆ~
1
i
ˆ
AB
C
ˆ ˆ,
A B 2 ,
(4.153)
2
2
and hence continue Eq. (148) as
ˆ~ ˆ~
1
A
B
AB
A ˆ B ˆ, ,
(4.154)
2
thus proving Eq. (140).
For the particular case of operators x ând p ˆ (or a similar pair of operators for another Cartesian x
coordinate), we may readily combine Eq. (140) with Eq. (2.14b) to prove the original Heisenberg’s uncertainty relation (2.13). For the spin-½ operators defined by Eq. (116)-(117), it is very simple (and highly recommended to the reader) to show that
3
3
Spin- ½:
ˆ
, ˆ
i
S S i S
(4.155)
j
j'
2
ˆ ,
jj'j"
j"
ˆ ˆ
i.e.
, j j'
ˆ ,
jj' j"
j"
commutation
j" 1
j" 1
relations
where jj’j” is the Levi-Civita permutation symbol.35 As a result, the uncertainty relations (140) for all Cartesian components of spin-½ systems are similar, for example
35 See, e.g., MA Eq. (13.2).
Chapter 4
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Spin-½:
uncertainty
S
S
S
etc
,
.
(4.156)
x
y
relations
2
z
In particular, as we already know, in the state the right-hand side of this relation equals (/2)2
> 0, so neither of the uncertainties Sx, Sy can equal zero. As a reminder, our direct calculation earlier in this section has shown that each of these uncertainties is equal to /2, i.e. their product is equal to the lowest value allowed by the uncertainty relation (156) – just as the Gaussian wave packets (2.16) provide the lowest possible value of the product x px, allowed by the Heisenberg relation (2.13).
4.6. Quantum dynamics: Three pictures
So far in this chapter, I shied away from the discussion of the system’s dynamics, implying that the bra- and ket-vectors were just their “snapshots” at a certain instant t. Now we are sufficiently prepared to examine their evolution in time. One of the most beautiful features of quantum mechanics is that this evolution may be described using either of three alternatives (called pictures), giving exactly the same final results for the expectation values of all observables.
From the standpoint of our wave-mechanics experience, the Schrödinger picture is the most natural one. In this picture, the operators corresponding to time-independent observables (e.g., to the Hamiltonian function H of an isolated system) are also constant in time, while the bra- and ket-vectors evolve in time as
( t) ( t ) ˆ†
u ( t, t ),
( t) ˆ u( t, t ) ( t ) .
(4.157a)
0
0
0
0
Here ˆ u( t, t ) is the time- evolution operator, which obeys the following differential equation: 0
i
ˆ
ˆ
u
ˆ u
H ,
(4.157b)
t
where H îs the Hamiltonian operator of the system – which is always Hermitian: H †
ˆ H ˆ , and t 0 is the
initial moment of time. (Note that Eqs. (157) remain valid even if the Hamiltonian depends on time explicitly.) Differentiating the second of Eqs. (157a) over time t, and then using Eq. (157b) twice, we can merge these two relations into a single equation, without explicit use of the time-evolution operator:
Schrödinger
i
equation
t
H ˆ
t ,
(4.158)
t
which is frequently more convenient. (However, for some purposes the notion of the time-evolution operator, together with Eq. (157b), are useful – as we will see in a minute.) While Eq. (158) is a very natural generalization of the wave-mechanical equation (1.25), and is also frequently called the Schrödinger equation,36 it still should be considered as a new, more general postulate, which finds its final justification (as it is usual in physics) in the agreement of its corollaries with experiment – more exactly, in the absence of a single credible contradiction to an experiment.
Starting the discussion of Eq. (158), let us first consider the case of a time-independent Hamiltonian, whose eigenstates an and eigenvalues En obey Eq. (68) for this operator:37
36 Moreover, we will be able to derive Eq. (1.25) from Eq. (158) – see below.
37 I have switched the state index notation from j to n, which was used for numbering stationary states in Chapter 1, to emphasize the special role played by the stationary states an in quantum dynamics.
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H ˆ a E a ,
(4.159)
n
n
n
and hence are also time-independent. (Similarly to the wavefunctions n defined by Eq. (1.60), an are called the stationary states of the system.) Let us use Eqs. (158)-(159) to calculate the law of time evolution of the expansion coefficients n (i.e. the probability amplitudes) defined by Eq. (118), in a stationary state basis, using Eq. (158):
d
d
1
E
i
ˆ
( t)
a ( t) a
( t) a
H ( t)
n
a ( t) E .
(4.160)
n
n
n
n
n
n
n
dt
dt
i
i
This is the same simple equation as Eq. (1.61), and its integration, with the initial moment t 0 taken for 0, yields a similar result – cf. Eq. (1.62):
Time
i
t
( ) ( )
0 exp
.
(4.161) evolution
n
n
E t
n
of probability
amplitudes
In order to illustrate how this result works, let us consider the dynamics of a spin-½ in a time-independent, uniform external magnetic field B. To construct the system’s Hamiltonian, we may apply the correspondence principle to the classical expression for the energy of a magnetic moment m in the external magnetic field B, 38
U m B .
(4.162)
In quantum mechanics, the operator corresponding to the moment m is given by Eq. (115) (suggested by W. Pauli), so the spin-field interaction is described by the so-called Pauli Hamiltonian, which may be, due to Eqs. (116)-(117), represented in several equivalent forms:
Pauli
ˆ
H m
ˆ
B Sˆ
B γ σˆ B .
(4.163a) Hamiltonian:
2
operator
If the z-axis is aligned with the field’s direction, this expression is reduced to H ˆ
B S ˆ
B ˆ .
(4.163b)
z
z
2
According to Eq. (117), in the z-basis of the spin states and , the matrix of the operator (163b) is
B
Ω
Pauli
H
σ σ ,
Ω
where
(4.164) Hamiltonian:
z
z
B
.
2
2
z-basis matrix
The constant so defined coincides with the classical frequency of the precession, about the z-axis, of an axially-symmetric rigid body (the so-called symmetric top), with an angular momentum S and the magnetic moment m = S, induced by the external torque = mB.39 (For an electron, with its negative gyromagnetic ratio e = – g e e/2 m e, neglecting the tiny difference of the g e-factor from 2, we get e
B ,
(4.165)
m e
so according to Eq. (3.48), the frequency coincides with the electron’s cyclotron frequency c.) 38 See, e.g., EM Eq. (5.100). As a reminder, we have already used this expression for the derivation of Eq. (3).
39 See, e.g., CM Sec. 4.5, in particular Eq. (4.72), and EM Sec. 5.5, in particular Eq. (5.114) and its discussion.
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In order to apply the general Eq. (161) to this case, we need to find the eigenstates an and eigenenergies En of our Hamiltonian. However, with our (smart :-) choice of the z-axis, the Hamiltonian matrix is already diagonal:
1
0
H
σ
,
(4.166)
2
z
2 0 1
meaning that the states and are the eigenstates of this system, with the eigenenergies, respectively, 40
Spin-½ in
magnetic
field:
E
and E .
(4.167)
eigenenergies
2
2
Note that their difference,
Δ E E E
Ω
,
(4.168)
B
corresponds to the classical energy 2 m B of flipping a magnetic dipole with the moment’s magnitude m = /2, oriented along the direction of the field B. Note also that if the product B is positive, then
is negative, so E is negative, while E is positive. This is in agreement with the classical picture of a magnetic dipole m having negative potential energy when it is aligned with the external magnetic field B – see Eq. (162) again.
So, for the time evolution of the probability amplitudes of these states, Eq. (161) immediately yields the following expressions:
i
i
( t) ( )
0 exp
t
t
t
(4.169)
,
( )
( )
0 exp
,
2
2
allowing a ready calculation of the time evolution of the expectation values of any observable. In particular, we can calculate the expectation value of Sz as a function of time by applying Eq. (130) to the (arbitrary) time moment t:
S ( t)
( t) *
( t) ( t) *
( t)
( )
0 *
( )
0 ( )
0 *
( )
0 S ( )
0 . (4.170)
z
2
2
z
Thus the expectation value of the spin component parallel to the applied magnetic field remains constant in time, regardless of the initial state of the system. However, this is not true for the components perpendicular to the field. For example, Eq. (132), applied to the moment t, gives
S
t
( )
*
*
0 *
0
0 *
0
.
(4.171)
x
t t t t
i t
e
i t
e
2
2
Clearly, this expression describes sinusoidal oscillations with frequency (164). The amplitude and the phase of these oscillations depend on initial conditions. Indeed, solving Eqs. (132)-(133) for the probability amplitude products, we get the following relations:
t *
t S
*
,
,
(4.172)
x t
i S y t
t t S
x t
i S y t
valid for any time t. Plugging their values for t = 0 into Eq. (171), we get 40 So, spin-½ gives one more example of two-level systems whose discussion was started in Sec. 2.6. The fact that all quantum two-level systems are isomorphic (see Sec. 5.1) adds importance to our current discussion.
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1
S ( t)
i t
i
t
x
Sx 0 i Sy
1
0 e
Sx 0 i Sy 0 e
2
2
(4.173)
S
x 0cos
t
S y 0sin t .
An absolutely similar calculation using Eq. (133) gives
S ( t) S
(4.174)
y
y 0cos
t
Sx 0sin t .
These formulas show, for example, that if at moment t = 0 the spin’s state was , i.e. Sx(0) =
Sy(0) = 0, then the oscillation amplitudes of both “lateral” components of the spin vanish. On the other hand, if the spin was initially in the state →, i.e. had the definite, largest possible value of Sx equal to /2
(in classics, we would say “the spin-½ was oriented in the x-direction”), then both expectation values
Sx and Sy oscillate in time41 with this amplitude, and with the phase shift /2 between them.
So, the quantum-mechanical results for the expectation values of the Cartesian components of spin-½ are indistinguishable from the classical results for the precession, with the frequency = –B, 42
of a symmetric top with the angular momentum L of magnitude /2, about the field’s direction (our axis z), under the effect of an external torque = mB exerted by the field B on the magnetic moment m =
L. Note, however, that the classical language does not describe the large quantum-mechanical uncertainties of the components, obeying Eqs. (156), which are absent in the classical picture – at least when the precession starts from a definite orientation of the angular momentum vector.43
Recall also that at the stationary orbital motion of a particle, the component Lz of its angular momentum is always a multiple of – see, e.g., Eq. (3.139). As a result, the angular momentum of a spin-½ particle, with its stationary values Sz = /2, cannot be explained by the summation of orbital moments of its hypothetical components, i.e. by any internal rotation of the particle about its axis.
After this illustration, let us return to the discussion of the general Schrödinger equation (157b) and prove the following fascinating fact: it is possible to write the general solution of this operator equation. In the easiest case when the Hamiltonian is time-independent, this solution turns out to be an exact analog of Eq. (161),
i
i
ˆ u ( t, t ) ˆ
ˆ
u ( t , t ) exp H t t
H t t
(4.175)
0
0
0
0
êxp
0 .
To start its proof we should, first of all, understand what a function (in this particular case, the exponent) of an operator means. In the operator (and matrix) algebra, such nonlinear functions are defined by their Taylor expansions; in particular, Eq. (175) means that
41 This is one more (hopefully, redundant :-) illustration of the difference between the averaging over the statistical ensemble and that over time: in Eqs. (170), (173)-(174), and also in quite a few relations below, only the former averaging has been performed, so the results are still functions of time.
42 Note that according to this relation, the gyromagnetic ratio may be interpreted as the angular frequency of the spin precession in a unit magnetic field – hence the name. In particular, for electrons, e 1.7611011 s-1T-1; for protons, the ratio is much smaller, p g p e/2 m p 2.675108 s-1T-1 – mostly because of their larger mass m p, at a g-
factor of the same order as for the electron: g p 5.586. For heavier spin-½ particles, e.g., atomic nuclei with such spin, the values of are correspondingly smaller – e.g., 8.681106 s-1T-1 for the 57Fe nucleus.
43 If the initial conditions are random, the classical motion is stochastic even if its laws are deterministic.
Chapter 4
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1 i
k
ˆ
ˆ
ˆ
u ( t, t ) I
H t t
0
0
k 1
k !
(4.176)
1
2
3
ˆ
i ˆ
I
i
i
H
t t
H t t
H t t
0
1
2
2
1
ˆ
ˆ
(
)
3 (
)3 ...,
!
1
2!
0
3!
0
where ˆ 2
ˆ ˆ
ˆ 3
ˆ ˆ ˆ
H H
H , H
H
H
H
, etc. Working with such a series of operator products is not as hard as one
could imagine, due to their regular structure. For example, let us differentiate both sides of Eq. (176) over t, at constant t 0, at the last step using this equality again – that time, backward:
1 i
1
2
i
2
1
2
i
ˆ
ˆ
ˆ
ˆ
u( t, t 0ˆ
)
H H 2( t t )
3
H (
3 t t )2 ...
0
t
!
1
2!
0
3!
0
(4.177)
i
1
2
ˆ ˆ
i
H I H
i
i
t t
H t t
u
H t t
0
1
ˆ
ˆ 2
2
ˆ
(
)
...
ˆ ( , ),
!
1
2!
0
0
so the differential equation (158) is indeed satisfied. On the other hand, Eq. (175) also satisfies the initial condition
u ˆ t
( , t ) u ˆ † t
( , t I ˆ
)
(4.178)
0
0
0
0
that immediately follows from the definition (157a) of the evolution operator. Thus, Eq. (175) indeed gives the (unique) solution for the time evolution operator – in the Schrödinger picture.
Now let us allow the operator H ˆ to be a function of time, but with the condition that its “values”
(in fact, operators) at different instants commute with each other:
H ˆ ( t' H ˆ
), ( t" ) ,
0
t'
any
for
, t" .
(4.179)
(A good example is the Pauli Hamiltonian (4.163) for a spin in a classical magnetic field B even if it depends on time. Indeed, the spin operator Sˆ does not depend explicitly on time and hence commutes with itself as well as with the c-numbers B( t’) and B( t” ). Note, however, that a similar operator describing the effect of a classical position-independent force F( t) on the orbital motion of a particle, ˆ
H F( t) rˆ ,
(4.180)
F
may be deceiving: though it satisfies Eq. (179), this relation is invalid for the particle’s full Hamiltonian including its kinetic energy.) In this case, it is sufficient to replace, in all the above formulas, the product ˆ
H ( t t ) with the corresponding integral over time; in particular, Eq. (175) is generalized as 0
Evolution
t
i
operator: