qm fundamental concepts
TRANSCRIPT
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
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.
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}.
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.
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.
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)
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)
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)
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.
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.
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.
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.
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).
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).
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.
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)
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)
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.
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.
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.
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)
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)
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.
30 Chapter 1. Fundamental concepts