24
Chapter 1 Fundamental concepts 1.1 The Stern-Gerlach experiment The Stern-Gerlach experiment is described in almost every text on quan- tum mechanics, including Section 1.1 of Sakurai and Napolitano. The sig- nificant features of the Stern-Gerlach experiment that are relevant to our considerations of quantum mechanics are Measurement of the projection of the magnetic moment of silver atoms in a fixed direction revealed that the distribution of measure- ments is wholly parallel or anti-parallel to that direction, rather than characteristic of a continuous distribution. The measurement forces the system into a particular state; only two such 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 the x- 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 explain these microscopic quantum phenomena. Quantum mechanics has been developed in various forms: wave mechanics (Schr ¨ odinger), ”matrix me- chanics” (Heisenberg), the ”symbolic method” (Dirac) and in a ”space- time” formalism (Feynman). In this course we consider Dirac’s formula- 7

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Page 1: QM Fundamental Concepts

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 silver

atoms 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 Dirac’s formula-

7

Page 2: QM Fundamental Concepts

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 |α〉 (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 |α〉,∣∣∣β⟩ ∈ V then there exists |α〉 +

∣∣∣β⟩ ∈ V (closure).

2. If |α〉 ∈ V and c ∈ C then c |α〉 ∈ V.

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

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

∣∣∣β⟩,∣∣∣γ

⟩ ∈ V we have

5. |α〉 +∣∣∣β⟩=

∣∣∣β⟩+ |α〉 (commutativity).

6.(

|α〉 +∣∣∣β⟩)

+

∣∣∣γ

⟩= |α〉 +

(∣∣∣β⟩+

∣∣∣γ

⟩)

(associativity).

7. 1 |α〉 = |α〉.

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

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

10. c1(

|α〉 +∣∣∣β⟩)

= c1 |α〉 + c1∣∣∣β⟩(distributivity). (1.2.1)

⊲ :The kets |α〉 and c |α〉 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 |α〉 ∈ V to giveA |α〉 ∈ V.

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1.2 Kets, bras and operators 9

⊲ : There are particular |α〉 ∈ 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;±

=1√2|Sz;+〉 ±

i√2|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{(

|α〉 ,∣∣∣β⟩)

, |α〉 ,∣∣∣β⟩ ∈ V

}

to the element denoted⟨α∣∣∣ β

⟩in C (the scalar

product) with the following properties:

1. If |α〉 ,∣∣∣β⟩ ∈ V then

⟨α∣∣∣ β

⟩=

⟨β∣∣∣ α

⟩⋆(⋆ denotes complex conjugation).

2. If |α〉 ∈ V then 〈α| α〉 ≥ 0 (a positive-definite metric).

3. If |α〉 ∈ V, then 〈α| α〉 = 0 if and only if |α〉 = 0. (1.2.5)

4. If |α〉 ,∣∣∣β⟩,∣∣∣γ

⟩ ∈ V and c1, c2 ∈ C then(

〈c1α| +⟨c2β

∣∣∣

) ∣∣∣γ

⟩= c1

⟨α∣∣∣ γ

⟩+ c2

⟨β∣∣∣ γ

⟩.

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

Page 4: QM Fundamental Concepts

10 Chapter 1. Fundamental concepts

⊲ : Two kets |α〉 and∣∣∣β⟩are said to be orthogonal if

⟨α∣∣∣ β

⟩= 0. (1.2.6)

⊲ : Given a ket |α〉 , |0〉 we can form a normalized ket |α〉:

|α〉 = 1√〈α| α〉

|α〉 (1.2.7)

with the property that 〈α| α〉 = 1, and√〈α| α〉 is the norm of |α〉†

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 |α〉 = Y |α〉 for all |α〉 ∈ V.

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

3. Operators can be added, and

X + Y = Y + X (commutative)

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

4. Generally speaking X(

c1 |α〉 + c2∣∣∣β⟩)

= c1X |α〉 + c2X∣∣∣β⟩, 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 |α〉) = (XY) |α〉 = XY |α〉 . (1.2.8)

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

〈α|(

A†∣∣∣β⟩)

= (〈α|A)∣∣∣β⟩. (1.2.9)

⊲ : An operator A isHermitian if A = A† and

〈α|A∣∣∣β⟩= (〈α|A)

∣∣∣β⟩=

⟨β∣∣∣A |α〉⋆ . (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.

Page 5: QM Fundamental Concepts

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′〉 = δa′′,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.

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12 Chapter 1. Fundamental concepts

1.3.1 Eigenkets as base kets

Given an arbitrary ket |α〉 we write

|α〉 =∑

a′

ca′ |a′〉 . (1.3.3)

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

ca′′ = 〈a′′| α〉 . (1.3.4)

So we may write

|α〉 =∑

a′

|a′〉 〈a′| α〉 (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,

〈α| α〉 = 〈α|

a′

|a′〉 〈a′|

|α〉 =

a′

|〈a′| α〉|2 (1.3.7)

from which it follows that if |α〉 is normalized then

a′

|ca′ |2 =∑

a′

|〈a′| α〉|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)

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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∣∣∣γ

⟩= X |α〉 then

⟨a′∣∣∣ γ

⟩= 〈a′|X |α〉 =

a′′

〈a′|X |a′′〉 〈a′′| α〉

or

a(1)∣∣∣ γ

a(2)∣∣∣ γ

...

=

a(1)∣∣∣X

∣∣∣a(1)

⟩ ⟨

a(1)∣∣∣X

∣∣∣a(2)

· · ·⟨

a(2)∣∣∣X

∣∣∣a(1)

⟩ ⟨

a(2)∣∣∣X

∣∣∣a(2)

· · ·...

.... . .

a(1)∣∣∣ α

a(2)∣∣∣ α

...

. (1.3.14)

Let us now look at

⟨β∣∣∣ α

⟩=

a′

⟨β∣∣∣ a′

⟩ 〈a′| α〉 (1.3.15)

or

⟨β∣∣∣ α

⟩=

( ⟨

a(1)∣∣∣ β

⟩⋆ ⟨

a(2)∣∣∣ β

⟩⋆· · ·

)

a(1)∣∣∣ α

a(2)∣∣∣ α

...

. (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′′〉 a′δa′′,a′ 〈a′| =∑

a′

a′ |a′〉 〈a′| . (1.3.17)

Page 8: QM Fundamental Concepts

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− |+〉 = ~ |−〉 . (1.3.20)

Furthermore if we let

|+〉 =(

10

)

, |−〉 =(

01

)

(1.3.21)

then we see the matrix representation of the operators:

Sz =~

2

(

1 00 −1

)

, S+ = ~

(

0 10 0

)

, S− = ~

(

0 01 0

)

(1.3.22)

.

For the operators, it is conventional to label the column (row) indicesin descending order of angular momentum components.

1.4 Measurements, observables and the uncer-

tainty relation

Before the measurement of observable A, the system is assumed to bein some linear combination

|α〉 =∑

a′

ca′ |a′〉 =∑

a′

|a′〉 〈a′| α〉 . (1.4.1)

⊲ :When a measurement is performed, the system is ‘thrown into’ one of

the eigenstates of A:

|α〉 measurement−→ |a′〉 , (1.4.2)

Page 9: QM Fundamental Concepts

1.4 Measurements, observables and the uncertainty relation 15

thus a measurement usually changes the state, except if |α〉 is |a′〉, aneigenstate. Then

|α〉 measurement−→ |a′〉 . (1.4.3)

⊲ :The probability of jumping into some particular |a′〉 is

Prob(a′) = |〈a′| α〉|2 (1.4.4)

assuming that |α〉 is normalized.

Note that due to orthogonality, the probability of |a′〉 measurement−→ |a′′〉 is zero.

⊲ : The expectation value‖ of A with respect to the state |α〉 is

〈A〉 = 〈α|A |α〉 . (1.4.5)

This agrees with our intuitive idea of an average measured value:

〈A〉 =∑

a′

a′′

〈α| a′′〉 〈a′′|A |a′〉 〈a′| α〉

=

a′

a′′

〈α| a′′〉 a′ 〈a′′| a′〉 〈a′| α〉

=

a′

a′︸︷︷︸

measured value

|〈a′| α〉|2︸ ︷︷ ︸

Prob(a′)

. (1.4.6)

⊲ (1 ·4 ·1) A -12.

Consider a positively polarized beam in the x direction with apparatusselecting the z component of spin. The probability that |Sx;+〉 is throwninto |Sz;±〉 is 1/2 for each. Then by P VI:

|〈+| Sx;+〉| = |〈−| Sx;+〉| = 1√2. (1.4.7)

We can therefore write by (1.4.1) before a measurement:

|Sx;+〉 = 1√2|+〉 + 1√

2eiδ1 |−〉 (1.4.8)

‖Note: do not confuse eigenvalues with expectation values.

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16 Chapter 1. Fundamental concepts

where δ1 is a real phase which does not change. Since |Sx;+〉 and |Sx;−〉 aremutually exclusive, orthogonality gives us

|Sx;−〉 = 1√2|+〉 − 1√

2eiδ1 |−〉 (1.4.9)

which we might check by evaluating 〈Sx;−| Sx;+〉 = 0.Using (1.3.17) we can construct

Sx =~

2

[(

|Sx;+〉 〈Sx;+|)

−(

|Sx;−〉 〈Sx;−|)]

=~

2

[

e−iδ1(

|+〉 〈−|)

+ eiδ1(

|−〉 〈+|)]

(1.4.10)

and similarly (but using a different relative phase as that for Sx):

∣∣∣Sy;±

=1√2|+〉 ± 1√

2eiδ2 |−〉 (1.4.11)

gives the operator in z basis:

Sy =~

2

[

e−iδ2(

|+〉 〈−|)

+ eiδ2(

|−〉 〈+|)]

. (1.4.12)

Let us now consider a positively polarized beam in the x direction withapparatus selecting only the y component of spin. Then, since we expectthe system to be invariant under rotations, we obtain by analogy with(1.4.7):

∣∣∣∣

Sy;±∣∣∣ Sx;+

⟩∣∣∣∣ =

∣∣∣∣

Sy;±∣∣∣ Sx;−

⟩∣∣∣∣ =

1√2. (1.4.13)

Using (1.4.8) and the first of (1.4.12) in (1.4.13), we obtain

12

∣∣∣1 ± ei(δ1−δ2)

∣∣∣ =

1√2

(1.4.14)

which is satisfied if

δ2 − δ1 = π/2 or − π/2 (1.4.15)

Note: the expectation value for a spin-12system can assume any real values

between −~/2 and ~/2. The eigenvalues of Sz assumes only two values:−~/2 and ~/2.

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1.4 Measurements, observables and the uncertainty relation 17

1.4.1 Compatible observables

⊲ : The observables A and B are compatible if [A,B] = 0, andincompatible if [A,B] = 0.

⊲ (1 ·4 ·2) The spin components S2 = S2x + S2

y + S2z and Sz are com-

patible. Sx and Sy are not.

So far, we have bypassed the issue of degeneracy. Is the space spannedby {|a′〉} complete if these are degenerate eigenvalues? Fortunately, inpractice there is usually some other commuting observable which can beused to label the degenerate eigenvalue.

⊲ (1 ·4 ·1) E .Suppose A and B are compatible observables, and that the eigenvalues ofA are nondegenerate. Then 〈a′′|B |a′〉 are only non-zero on the diagonal.

⊲ (1 ·4 ·1) A and B are compatible, so

〈a′′| [A,B] |a′〉 = 0 → 〈a′′|AB |a′〉 − 〈a′′|BA |a′〉 = 0

Evaluating the operator A to the appropriate side using (1.2.10),

a′′ 〈a′′|B |a′〉 − a′ 〈a′′|B |a′〉 = (a′′ − a′) 〈a′′|B |a′〉 = 0

therefore each matrix measurement must satisfy

〈a′′|B |a′〉 = δa′,a′′ 〈a′|B |a′〉 . (1.4.16)

Using (1.3.9) and (1.4.16), we can write any operator B as

B =∑

a′′

|a′′〉 〈a′′|B |a′′〉 〈a′′| . (1.4.17)

Suppose that B operates on an eigenket of A:

B |a′〉 =∑

a′′

|a′′〉 〈a′′|B |a′′〉 〈a′′| a′〉

= |a′〉 〈a′|B |a′〉 〈a′| a′〉= 〈a′|B |a′〉 |a′〉 ≡ b′ |a′〉 (1.4.18)

where we identify 〈a′|B |a′〉 |a′〉 as b′. Therefore |a′〉 is a simultaneous eigen-ket of A and B.

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18 Chapter 1. Fundamental concepts

The statement that compatible observables have simultaneous eigen-kets also holds if there is an n-fold degeneracy, that is

A∣∣∣a′(i)

= a′∣∣∣a′(i)

for i = 1, 2, . . . , n (1.4.19)

where the∣∣∣a′(i)

are mutually orthonormal. To see this we need to construct

appropriate linear combinations of∣∣∣a′(i)

that diagonalize B, following the

diagonalization procedure discussed in Section 1.5 of Sakurai. A remarkon notation: the simultaneous eigenkets of A and B are denoted by |a′, b′〉or sometimes just by |k′〉 = |a′, b′〉. The order in which one measurescompatible observables does not matter.

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1.4 Measurements, observables and the uncertainty relation 19

1.4.2 The uncertainty relation

⊲ : Given an observable A, we define the operator

∆A = A − 〈A〉 . (1.4.20)

Then the quantity

(∆A)2⟩

=

A2 − 2A 〈A〉 + 〈A〉2⟩

=

A2⟩

− 〈A〉2 (1.4.21)

is called the variance (or mean square deviation or dispersion) of A.

⊲ (1 ·4 ·3) T |Sz;+〉 -12 .To calculate the variance of of Sz and Sx operator in this state,

(∆Sz)2⟩

=

S2z

−⟨

Sz

⟩2=

(

~

2

)2

−(

~

2

)2

= 0

(∆Sx)2⟩

=

S2x

−⟨

Sx

⟩2=~2

4.

Sz is ‘sharp’ while Sx is ‘fuzzy’.

⊲ (1 ·4 ·2) T S .

〈α| α〉 ⟨β∣∣∣ β

⟩ ≥∣∣∣⟨α∣∣∣ β

⟩∣∣∣2

(1.4.22)

⊲ (1 ·4 ·2) First note that for any λ ∈ Cwe must have

(

〈α| + λ⋆ ⟨β∣∣∣

)

·(

|α〉 + λ∣∣∣β⟩) ≥ 0

(The λ⋆ must be there to satisfy the inner product space condition⟨

α∣∣∣ β

=⟨β∣∣∣ α

⟩⋆, and 〈α| α〉 ≥ 0must also be satisfied.) Choosing λ = − ⟨

β∣∣∣ α

⟩/⟨β∣∣∣ β

gives us the result we want.

⊲ (1 ·4 ·3) H .The expectation value of an Hermitian operator is real.

⊲ (1 ·4 ·3) Follows trivially from (1.2.10).

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20 Chapter 1. Fundamental concepts

⊲ (1 ·4 ·4) A-H .The expectation value of an anti-Hermitian operator, defined by C = −C†,is purely imaginary.

⊲ (1 ·4 ·4) Also trivial.

Using these lemmas, we can now prove the uncertainty relations.

⊲ (1 ·4 ·5) For any two observables A and B, we can say that

(∆A)2⟩ ⟨

(∆B)2⟩

≥ 1

4

∣∣∣〈[A,B]〉

∣∣∣2. (1.4.23)

⊲ (1 ·4 ·5) Let |α〉 = ∆A∣∣∣γ

⟩and

∣∣∣β⟩= ∆B

∣∣∣γ

⟩, where

∣∣∣γ

⟩is any ket. Then

from Lemma (1.4.2):

(⟨γ∣∣∣∆A

)

∆A∣∣∣γ

⟩ (⟨γ∣∣∣∆B

)

∆B∣∣∣γ

⟩ ≥∣∣∣

(⟨γ∣∣∣∆A

)

∆B∣∣∣γ

⟩∣∣∣2.

Using the fact that ∆A and ∆B are Hermitian, this becomes

(∆A)2⟩ ⟨

(∆B)2⟩

≥∣∣∣〈∆A ∆B〉

∣∣∣2.

Note that ∆A ∆B = 12[∆A,∆B] + 1

2{∆A,∆B}, where the second expression

is an anticommutator, ∆A ∆B + ∆B ∆A.

Now [∆A,∆B] = [A,B] and is anti-Hermitian while {∆A,∆B} is Hermi-tian, so using Lemmas (1.4.3) and (1.4.4):

〈∆A ∆B〉 = 12〈[A,B]〉︸ ︷︷ ︸

imaginary

+12〈{∆A,∆B}〉︸ ︷︷ ︸

real

which finally gives us

∣∣∣∆A ∆B

∣∣∣2=

1

4

∣∣∣〈[A,B]〉

∣∣∣2+1

4

∣∣∣〈{A,B}〉

∣∣∣2.

This is even stronger than the traditional statement in (1.4.23).

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1.5 Change of basis 21

1.5 Change of basis

Suppose we have two incompatible observables, A and B. The vectorspace can be viewed as spanned either by the set {|a′〉} or the set {|b′〉}.

⊲ (1 ·5 ·1) A -12.

The states |Sz;±〉may be used as our base kets. A viable alternative wouldbe the states |Sx;±〉.

How are these two descriptions related? We need to construct a trans-formation that connects {|a′〉} and {|b′〉}.

⊲ (1 ·5 ·1) B .Given two sets of base kets, both satisfying orthonormality and complete-ness, there exists a unitary operatorU∗∗ such that

∣∣∣b(1)

=U∣∣∣a(1)

,∣∣∣b(2)

=U∣∣∣a(2)

, . . . ,∣∣∣b(N)

=U∣∣∣a(N)

. (1.5.1)

⊲ (1 ·5 ·1) Rather than derive the operator from scratch, we just let

U =∑

k

∣∣∣b(k)

⟩ ⟨

a(k)∣∣∣ . (1.5.2)

Then

U∣∣∣a(l)

=

k

∣∣∣b(k)

⟩ ⟨

a(k)∣∣∣ a(l)

=

∣∣∣b(l)

.

Furthermore,

U†U =∑

k

l

∣∣∣a(l)

⟩ ⟨

b(l)∣∣∣ b(k)

⟩ ⟨

a(k)∣∣∣ =

k

∣∣∣a(k)

⟩ ⟨

a(k)∣∣∣ = I,

and similarly,UU† = I.

Note that if we can describe in each of the bases |α〉 =∑

l

∣∣∣a(l)

⟩ ⟨

a(l)∣∣∣ α

=∑

k

∣∣∣b(k)

⟩ ⟨

b(k)∣∣∣ α

, then

b(k)∣∣∣ α

=

l

b(k)∣∣∣ a(l)

⟩ ⟨

a(l)∣∣∣ α

=

l

a(k)∣∣∣U†

∣∣∣a(l)

⟩ ⟨

a(l)∣∣∣ α

,

which takes the form of matrix multiplication:

∗∗By unitary, we mean to say thatU satisfiesUU† =U†U = I.

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22 Chapter 1. Fundamental concepts

=

U†

.

The relationship between matrix elements in the two bases:

b(k)∣∣∣X

∣∣∣b(l)

=

m

n

b(k)∣∣∣ a(m)

⟩ ⟨

a(m)∣∣∣X

∣∣∣a(n)

⟩ ⟨

a(n)∣∣∣ b(l)

=

m

n

a(k)∣∣∣U†

∣∣∣a(m)

⟩ ⟨

a(m)∣∣∣X

∣∣∣a(n)

⟩ ⟨

a(n)∣∣∣U

∣∣∣a(l)

therefore⟨

b(k)∣∣∣X

∣∣∣b(l)

=

a(k)∣∣∣X′

∣∣∣a(l)

, where X′ = U†XU.

1: Show that tr(X) =∑

a′ 〈a′|X |a′〉 is independent of the basis andthat tr(XY) = tr(YX).

Wemaywish to diagonalize thematrix representation of an operator B.It allows us to find eigenvalues and eigenkets of B, given the set {〈a′′|B |a′〉}.

B |b′〉 = b′ |b′〉 or∑

a′

〈a′′|B |a′〉 〈a′| b′〉 = b′ 〈a′′| b′〉 . (1.5.3)

The above has the form of a matrix eigenvalue problem. The Hermiticityof B is important. The operator S+ defined in (1.3.22) is non-Hermitian, soit cannot be diagonalized by any unitary matrix.

⊲ (1 ·5 ·2) C .Consider two sets of orthonormal bases {|a′〉} and {|b′〉} connected by theoperator U in (1.5.2). Construct the unitary transform UAU−1 of A. The|b′〉’s are eigenkets ofUAU−1 with exactly the same eigenvalues as A.

⊲ (1 ·5 ·2) We start with a statement of the eigenvalue problem, where

the solutions are known in{∣∣∣a(l)

⟩}

:

A∣∣∣a(l)

= a(l)∣∣∣a(l)

then then multiply on the left by U and insert an identity statementU−1U = I on the left hand side:

UAU−1U∣∣∣a(l)

= a(l)U∣∣∣a(l)

and finally note thatU∣∣∣a(l)

=

∣∣∣b(l)

as in (1.5.2):

(

UAU−1) ∣∣∣b(l)

= a(l)∣∣∣b(l)

. (1.5.4)

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1.6 Position, momentum and translation 23

We note thatUAU−1 is the same as B itself only ifU−1 = U†, so we requirethatU be unitary.

⊲ (1 ·5 ·2) A -12.

The quantities Sx and Sz are related by a unitary operatorwhich is a rotationabout the y-axis through π/2 (see Chapter 3 of Sakurai). Sx and Sz have thesame set of eigenvalues: +~/2 and −~/2. The theorem holds in this case.

1.6 Position, momentum and translation

For continuous spectra (eg. pz, the z-component of momentum), somegeneralizations are in order:

1. A |a′〉 = a′ |a′〉 −→ ξ |ξ′〉 = ξ′ |ξ′〉.

2. 〈a′| a′′〉 = δa′,a′′ −→ 〈ξ′| ξ′′〉 = δ(ξ′ − ξ′′).

3.∑

a′ |a′〉 〈a′| = I −→∫

dξ′ |ξ′〉 〈ξ′| = I.

4. |α〉 =∑

a′ |a′〉 〈a′| α〉 −→ |α〉 =∫

dξ′ |ξ′〉 〈ξ′| α〉.

5.∑

a′ |〈a′| α〉|2 = 1 −→∫

dξ′ |〈ξ′| α〉|2 = 1.

6.⟨β∣∣∣ α

⟩=

a′⟨β∣∣∣ a′

⟩ 〈a′| α〉 −→ ⟨β∣∣∣ α

⟩=

dξ′⟨β∣∣∣ ξ′

⟩ 〈ξ′| α〉.

7. 〈a′′|A |a′〉 = a′δa′,a′′ −→ 〈ξ′′| ξ |ξ′〉 = ξ′δ(ξ′′ − ξ′). (1.6.1)

1.6.1 Position

Consider the position operator x in one dimension:

x |x′〉 = x′ |x′〉 . (1.6.2)

⊲ :We assume that |α〉 forms a complete set.

We may write any |α〉 describing an arbitrary physical state as

|α〉 =∫ ∞

−∞dx′ |x′〉 〈x′| α〉 (1.6.3)

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24 Chapter 1. Fundamental concepts

Consider an idealized measurement where we have a very small detectorwhich clicksonlywhen theparticle is at x′ (andnowhere else). Immediatelyafter the detector clicks, we can say that the state of the system is |x′〉, i.e.,

|α〉 measurement−→ |x′〉 .In practice, the best we can do is to locate the particle in a narrow interval∆ around x′. In other words,

|α〉 =∫ ∞

−∞dx′′ |x′′〉 〈x′′| α〉 measurement−→

∫ x′+∆/2

x′−∆/2dx′′ |x′′〉 〈x′′| α〉 . (1.6.4)

Assuming that 〈x′′| α〉 does not vary much over ∆, the probability forthe detector to click is given by the continuous analog of P VI:

|〈x′| α〉|2 dx′ (1.6.5)

where we have written dx′ for ∆. The probability of recording the particlesomewhere between −∞ and∞ is given by

∫ ∞

−∞dx′ |〈x′| α〉|2 =

∫ ∞

−∞dx′ 〈α| x′〉 〈x′| α〉 dx′ = 〈α| α〉 = 1 (1.6.6)

assuming that |α〉 is normalized - also noting that 〈x′| α〉 is thewavefunctionfor state |α〉.

Generalizing to three dimensions:

|α〉 =∫

dx′ |x′〉 〈x′| α〉 (1.6.7)

where |x′〉 ≡∣∣∣x′y′z′

⟩is a simultaneous eigenket of the observables x, y and

z. We are implicitly assuming that

[

xi, x j

]

= 0 (1.6.8)

where x1, x2 and x3 stand for x, y and z respectively.

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1.6 Position, momentum and translation 25

1.6.2 Translation

⊲ : The operator for infinitesimal translation of a system local-ized at x′ by dx′ is

T (dx′) |x′〉 = |x′ + dx′〉 . (1.6.9)

If T (dx′) operates on an arbitrary state |α〉 we can write

|α〉 → T (dx′) |α〉 = T (dx′)∫

dx′ |x′〉 〈x′| α〉

=

dx′ |x′ + dx′〉 〈x′| α〉

=

dx′ |x′〉 〈x′ − dx′| α〉 (1.6.10)

therefore the wavefunction of the translated state T (dx′) |α〉 is obtained bysubstituting x′ − dx′ for x′ in 〈x′| α〉.The operator T (dx′) should have the following properties:

1. 〈α| α〉 = 〈α| T †(dx′)T (dx′) |α〉 = 1 is a reasonable assumption, assuredif T †(dx′)T (dx′) = I.

2. T (dx′′)T (dx′) = T (dx′ + dx′′)

3. T (−dx′) = T −1(dx′)

4. limdx′→0T (dx′) = I (1.6.11)

2: If we choose

T (dx′) = I − iK ·dx′ (1.6.12)

where K and its components Kx,Ky and Kz are Hermitian operators, thenshow that all the properties listed above are satisfied.

⊲ : The operator K is the generator of infinitesimal spatial trans-lations.

Page 20: QM Fundamental Concepts

26 Chapter 1. Fundamental concepts

We will give physical meaning to K by assuming that††:

p ≡ ~K (1.6.13)

where p is the linear momentum with units MLT−1, ~ is a fundamentalconstant with units ML2T−1, and K has units L−1.Then (1.6.12) may be written as

T(dx′) = I − ip ·dx′/~ (1.6.14)

So at this point we have two fundamental observables:

• the position x from the underlying spatial ‘arena’;

• the linear momentum p from translations in space.

Later on we will see two more observables:

• the HamiltonianH or total energy generated fromtime displacements;

• the angular momentum L from rotations in space.

(1.6.15)

Now let us ask the question: are x and p compatible?

We have

xT (dx′) |x′〉 = x |x′ + dx′〉 = (x′ + dx′) |x′ + dx′〉T (dx′)x |x′〉 = T (dx′)x′ |x′〉 = x′T (dx′) |x′〉 = x′ |x′ + dx′〉

and so the commutator of the two is

[x,T (dx′)] |x′〉 = dx′ |x′ + dx′〉= dx′

(I − ip ·dx′/~) |x′〉

≈ dx′ |x′〉

where in the second step, we use (1.6.9) and (1.6.14). So casting out thesecond order term, we have shown that

[x,T (dx′)] = dx′ (1.6.16)

††See Sakurai for a hand-waving justification.

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1.6 Position, momentum and translation 27

Using (1.6.14) in (1.6.16), we obtain

x(

I − i~p ·dx′

)

−(

I − i~p ·dx′

)

x = dx′

or − i~

(

x(p ·dx′) − (p ·dx′)x)

= dx′. (1.6.17)

For the special casewheredx′ = dx′x, the above simplifies to− i~dx′

(xpx − pxx

)=

dx and so

[x, px

]= i~δi j,

[

y, py]

= 0,[z, pz

]= 0. (1.6.18)

We can get similar relations for dx′ = dy′ y and dx′ = dz′z, hence

[

xi, p j

]

= i~δi j. (1.6.19)

Using (1.4.23) it follows that

(∆xi)2⟩ ⟨(∆pi

)2⟩

≥ 1

4

∣∣∣〈i~〉

∣∣∣2=~2

4or in a more familiar form,

δxi δpi ≥~

2, δxi =

〈(∆xi)2〉, δpi =

√⟨

(∆pi)2⟩

. (1.6.20)

However, unlike components can be simultaneously measured.

⊲ : A finite translation can be written as

T (∆x′) = limN→∞

(

1 − i~p · ∆x′

N

)N= exp

(

− i~p ·∆x′

)

. (1.6.21)

3: Check the steps in (1.6.21).

We must have

T (∆1x′)T (∆2x

′) = T (∆1x′+ ∆2x

′). (1.6.22)

Now the following also commutes:

eAeB = eA+B ⇔ [A,B] . (1.6.23)

4: Check the steps in (1.6.23).

So (1.6.21) and (1.6.23) imply that

[

pi, p j

]

= 0 ∀i, j. (1.6.24)

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28 Chapter 1. Fundamental concepts

1.7 Wavefunctions in position and momentum

space

Let us deduce how the operator p may look in the basis {|x′〉}.

(

I − i~p ·dx′

)

|α〉 =∫

dx′′T (dx′) |x′′〉 〈x′′| α〉

=

dx′′ |x′′ + dx′〉 〈x′′| α〉

=

dx′′ |x′′〉 〈x′′ − dx′| α〉

=

dx′′ |x′′〉(

〈x′′| α〉 − dx′ · ∇′′ 〈x′′| α〉)

(1.7.1)

where in the last step we have used a Taylor series expansion.This gives us

p |α〉 =∫

dx′(

−i~∇′ 〈x′| α〉)

(1.7.2)

or 〈x′| p |α〉 = −i~∇′ 〈x′| α〉 . (1.7.3)

From (1.7.2) we get

⟨β∣∣∣ p |α〉 =

dx′⟨β∣∣∣ x′

⟩ (−i~∇′ 〈x′| α〉)

≡∫

dx′ψ⋆β (x′)(

−i~∇′)

ψα(x′). (1.7.4)

Now in (1.7.3) we let |α〉 =∣∣∣p′

⟩:

〈x′|p∣∣∣p′

⟩= −i~∇′ ⟨x′

∣∣∣ p′

⟩(1.7.5)

or p′⟨x′∣∣∣ p′

⟩= −i~∇′ ⟨x′

∣∣∣ p′

⟩. (1.7.6)

The solution to this differential equation for⟨x′∣∣∣ p′

⟩is

⟨x′∣∣∣ p′

⟩= N exp

(i~p′ · x′

)

(1.7.7)

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1.7 Wavefunctions in position and momentum space 29

where N is a normalization constant.Using continuous orthonormality (〈ξ′| ξ′′〉 = δ(ξ′ − ξ′′)) we can write

δ(x′ − x′′) = 〈x′| x′′〉 =∫

dp′⟨

x′∣∣∣ p′

⟩ ⟨

p′∣∣∣ x′′

= |N|2∫

dp′ exp[i~p′ · (x′ − x′′)

]

= (2π~)3 |N|2 δ(x′ − x′′) (1.7.8)

where we have used (1.7.7) in the second step, and the following Dirac-delta properties in the last:

dp exp(i2πx ·p) = δ(x) and δ(ax) = 1

|a|3δ(x)

Then we may write the solution as

⟨x′∣∣∣ p′

⟩=

1

(2π~)3/2exp

(i~p′ · x′

)

(1.7.9)

which gives us

〈x′| α〉 ≡ ψα(x′) =∫

dp′⟨x′∣∣∣ p′

⟩ ⟨p′

∣∣∣ α

=1

(2π~)3/2

dp′ exp(i~p′ · x′

)

ψα(p′) (1.7.10)

or⟨p′

∣∣∣ α

⟩ ≡ ψα(p′) =∫

dx′⟨p′

∣∣∣ x′

⟩ 〈x′| α〉

=1

(2π~)3/2

dx′ exp(

− i~p′ · x′

)

ψα(x′). (1.7.11)

So real and momentum space wavefunctions are a Fourier transform pair.

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30 Chapter 1. Fundamental concepts