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• Chapter 1

Fundamental concepts

1.1 The Stern-Gerlach experiment

The Stern-Gerlach experiment is described in almost every text onquan-tum mechanics, including Section 1.1 of Sakurai and Napolitano. The sig-nificant features of the Stern-Gerlach experiment that are relevant to ourconsiderations of quantum mechanics are

Measurement of the projection of the magnetic moment of silveratoms in a fixed direction revealed that the distribution of measure-ments is wholly parallel or anti-parallel to that direction, rather thancharacteristic of a continuous distribution.

The measurement forces the system into a particular state; only twosuch states are accessible in the classic Stern-Gerlach experiment (la-belled spin-up and spin-down).

Repeated applications of the Stern-Gerlach experiment cause the sys-tem to lose all recollection of previous measurements, in this case thex- and y-components of the magnetic moment.

A quantum theory of measurement is required to explain these phe-nomena.

These observations constrain the form of acceptable theories to explainthese microscopic quantum phenomena. Quantum mechanics has beendeveloped in various forms: wave mechanics (Schrodinger), matrix me-chanics (Heisenberg), the symbolic method (Dirac) and in a space-time formalism (Feynman). In this course we consider Diracs formula-

7

• 8 Chapter 1. Fundamental concepts

tion which emphasises the superposition principle and the specification ofcomplex vector space representations of the states of a given system.

1.2 Kets, bras and operators

:The state of a physical system is represented by a state vector | (Dirac

notation - a ket) in a complex vector space V. (Complex denotes V isdefined over the field of complex numbers C.)

Properties of the complex vector spaceV:

1. If |, V then there exists | +

V (closure).

2. If | V and c C then c | V.

3. There exists |0 V such that | + |0 = | for all | V (null ket).

4. If | V then there is an inverse, |, such that | + ( |) = |0.For all | ,

, V we have

5. | +=

+ | (commutativity).

6.(

| +)

+

= | +

(+

)

(associativity).

7. 1 | = |.

8. c1 (c2 |) = (c1c2) | (associativity).

9. (c1 + c2) | = c1 | + c2 | (distributivity).

10. c1(

| +)

= c1 | + c1(distributivity). (1.2.1)

:The kets | and c | with c , 0 represent the same physical state.

:An observable of the physical system (eg. momentum or components of

spin) is represented by an operator A which operates on | V to giveA | V.

• 1.2 Kets, bras and operators 9

: There are particular | V which are the eigenkets of Adenoted |a , |a , . . . such that

A |a = a |a , A |a = a |a , . . . (1.2.2)

where a, a C and are called the eigenvalues of A.

:When a measurement is performed, the result is always an eigenvalue ofA, which suggests that A is such that the eigenvalues are all real.

: The physical state of the system corresponding to a particulareigenvalue (and eigenstate) is called an eigenstate.

(1 2 1) T -12.

Sz |Sz;+ =~

2|Sz;+ and Sz |Sz; =

~

2|Sz; (1.2.3)

The dimensionality of the space V is determined by the degrees of free-dom (two in this example). Any vector in the space can bewritten in termsof the eigenkets of a particular observable, for instance

Sy;

=12|Sz;+

i2|Sz; . (1.2.4)

We now introduce some additional requirements on the vector spaceV. We require that it is an inner product space. That means there exists amapping from the Cartesian product ofVwith itself, or the set of ordered

pairs{(

| ,)

, | , V

}

to the element denoted

in C (the scalar

product) with the following properties:

1. If | , V then

=

( denotes complex conjugation).

2. If | V then | 0 (a positive-definite metric).

3. If | V, then | = 0 if and only if | = 0. (1.2.5)

4. If | ,, V and c1, c2 C then

(

c1| +c2

) = c1

+ c2

.

The Cartesian product of setsA and B is defined as the setAB = {(a, b) : a A, b B}.

• 10 Chapter 1. Fundamental concepts

: Two kets | andare said to be orthogonal if

= 0. (1.2.6)

: Given a ket | , |0 we can form a normalized ket |:

| = 1|

| (1.2.7)

with the property that | = 1, and| is the norm of |

Let us now say more about the properties of the operators on V (ingeneral, not necessarily those corresponding to observables):

1. X = Y if and only if X | = Y | for all | V.

2. X is the null operator if and only if X | = 0 for all | V.

3. Operators can be added, and

X + Y = Y + X (commutative)

X + (Y + Z) = (X + Y) + Z (associative).

4. Generally speaking X(

c1 | + c2)

= c1X | + c2X, but not for

the case of the time reversal operator in Chapter 4 of Sakurai.

5. Operators can be multiplied -

XY , YX in general (non-commutative)

X(YZ) = (XY)Z = XYZ (associative)

X (Y |) = (XY) | = XY | . (1.2.8)

: The adjoint of an operator A can be defined to be the operatorA such that

|(

A)

= (|A). (1.2.9)

: An operator A isHermitian if A = A and

|A= (|A)

=A | . (1.2.10)

Note: if there is a norm defined on the inner product space then it is called a Hilbertspace, although some would only do so if the space is one of infinite dimension.

• 1.3 Base kets and matrix representations 11

1.3 Base kets and matrix representations

(1 3 1) Hermitian operators have three properties of extremeimportance in quantum mechanics:

1. The eigenvalues of an Hermitian operator are real.

2. The eigenfunctions of an Hermitian operator are orthogonal.

3. The eigenfunctions of an Hermitian operator form a complete set.

(1 3 1) We have A |a = a |a and A |a = a |a, so passingthrough the right and left respectively we get

a|A |a = a a| a and a|A |a = a a| a

a a| a = a a| a (a a) a| a = 0.

Now a and a can be the same or different. If they are the same then(a a) a| a = 0 a = a (assuming that |a , |0). Let us nowassume that a and a are different. Then a a = a a, which cannotbe zero by assumption, so

a| a = 0 (a , a), (1.3.1)

which proves orthogonality.

We can orthonormalize to form a complete set:

a| a = a,a . (1.3.2)

For the argument of completeness, note that we have implicitly as-sumed that the whole vector space is spanned by the eigenkets of A. Thisissue can be studied more rigorously using Sturm-Liouville theory.

If we require that the operators corresponding to observables are Her-mitian then they will have real eigenvalues, the importance of which willbecome clearer in the next section.

Linearity is also necessary.The possibility of a degenerate state has been ignored!So in P IV the operators should be Hermitian.

• 12 Chapter 1. Fundamental concepts

1.3.1 Eigenkets as base kets

Given an arbitrary ket | we write

| =

a

ca |a . (1.3.3)

Multiplying through from the left by a| and using the orthonormalityproperty (1.3.1) we find

ca = a| . (1.3.4)So we may write

| =

a

|a a| (1.3.5)

from which we can infer that

a

|a a| = I (completeness/closure relation) (1.3.6)

where I is the unity operator. This is a very useful expression of I. (1 3 1) T .

Inserting a completeness relation,

| = |

a

|a a|

| =

a

|a| |2 (1.3.7)

from which it follows that if | is normalized then

a

|ca |2 =

a

|a| |2 = 1. (1.3.8)

1.3.2 Matrix representations

For an operator X we may write

X =

a

a

|a a|X |a a| . (1.3.9)

Assuming an N-dimensional vector space, there are N2 numbers of theform

row a|X |a column (1.3.10)

• 1.3 Base kets and matrix representations 13

which we can write explicitly in the matrix form as follows:

X =

a(1)X

a(1)

a(1)X

a(2)

a(2)X

a(1)

a(2)X

a(2)

...

.... . .

. (1.3.11)

Referring back to Equation (1.2.10) we see that if X is Hermitian then

a|X |a = a|X |a (1.3.12)which is a property of an Hermitian matrix.

If Z = XY then

a|Z |a = a|XY |a =

a

a|X |a a|Y |a (1.3.13)

the standard way of multiplying two matrices.

If= X | then

a

= a|X | =

a

a|X |a a|

or

a(1)

a(2)

...

=

a(1)X

a(1)

a(1)X

a(2)

a(2)X

a(1)

a(2)X

a(2)

...

.... . .

a(1)

a(2)

...

. (1.3.14)

Let us now look at

=

a

a

a| (1.3.15)

or

=

(

a(1)

a(2)

)

a(1)

a(2)

...

. (1.3.16)

The matrix representation of an observable (operator) A becomes simple ifthe eigenkets of A are used as the base kets:

A =

a

a

|a a|A |a a| =

a

|a aa,a a| =

a

a |a a| . (1.3.17)

• 14 Chapter 1. Fundamental concepts

(1 3 2) A -12.

We have as base kets |Sz;, and use here for brevity |.The identity operator is I = |+ +| + | |, using Equation (1.3.6).Then by Equation (1.3.17), we have operator

Sz =~

2

[

(|+ +|) (| |)]

(1.3.18)

and we note also that Sz | = (~

2

)

|.Define the operators

S+ ~ |+ | and S ~ | +| (1.3.19)

which raise (S+) or lower (S) the spin component if possible, i.e.,

S+ | = ~ |+ , S+ |+ = 0, S | = 0, S |+ = ~ | . (1.3.