Linear Vector Spaces and Hilbert Space in Advanced Quantum Mechanics, Study notes of Quantum Mechanics

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PHY

Advanced Quantum Mechanics

Pieter Kok, The University of Sheffield.

August 2015

PHY472: Lecture Topics

1. Linear vector spaces
2. Operators and the spectral decomposition
3. Observables, projectors and time evolution
4. Tensor product spaces
5. The postulates of quantum mechanics I
6. The postulates of quantum mechanics II
7. The Schrödinger and Heisenberg picture
8. Mixed states and the density matrix
9. Perfect and imperfect measurements
10. Composite systems and entanglement
11. Quantum teleportation
12. Open quantum systems and completely positive maps
13. Orbital angular momentum
14. Spin angular momentum
15. Total angular momentum
16. Identical particles
17. Bose-Einstein and Fermi-Dirac statistics
18. Spin waves in solids
19. An atom in a cavity
20. Quantum field theory
21. Revision lecture (week before the exam)

PHY472: Advanced Quantum Mechanics

• 1 Linear Vector Spaces and Hilbert Space Contents
  • 1.1 Linear vector spaces
  • 1.2 Operators in Hilbert space
  • 1.3 Hermitian and unitary operators
  • 1.4 Projection operators and tensor products
  • 1.5 The trace and determinant of an operator
• 2 The Postulates of Quantum Mechanics
• 3 Schrödinger and Heisenberg Pictures
• 4 Mixed States and the Density Operator
  • 4.1 Mixed states
  • 4.2 Decoherence
  • 4.3 Imperfect measurements
• 5 Composite Systems and Entanglement
  • 5.1 Composite systems
  • 5.2 Entanglement
  • 5.3 Quantum teleportation
• 6 Evolution of Open Quantum Systems
  • 6.1 The Lindblad equation
  • 6.2 Positive and completely positive maps
• 7 Orbital Angular Momentum and Spin
  • 7.1 Orbital angular momentum
  • 7.2 Spin
  • 7.3 Total angular momentum
  • 7.4 Composite systems with angular momentum
• 8 Identical Particles
  • 8.1 Symmetric and anti-symmetric states
  • 8.2 Creation and annihilation operators
  • 8.3 Observables based on creation and annihilation operators
  • 8.4 Bose-Einstein and Fermi-Dirac statistics
• 9 Many-Body Problems in Quantum Mechanics
  • 9.1 Interacting electrons in atomic shells
  • 9.2 Spin waves in solids
  • 9.3 An atom in a cavity
  • 9.4 Outlook: quantum field theory

6 PHY472: Advanced Quantum Mechanics

Suggested Further Reading

1. Quantum Processes, Systems, and Information , by Schumacher and Westmoreland,

Cambridge University Press (2010). This is an excellent book, and should be your first

choice for additional material. It has everything up to many-body quantum mechanics.

2. Quantum Information and Quantum Computation , by Nielsen and Chuang, Cam-

bridge University Press (2000). This is the current standard work on quantum information

theory. It has a comprehensive introduction to quantum mechanics along the lines treated

here, but in more depth. The book is from 2000, which means that several important recent

topics are not covered.

3. Introductory Quantum Optics , by Gerry and Knight, Cambridge University Press (2005).

This is a very accessible introduction to the quantum theory of light.

4. Quantum Field Theory , by Lewis Ryder, Cambridge University Press (1996). This is a

quite advanced introduction to relativistic quantum field theory.

I would like to thank the students who have used these lecture notes in previous years and spotted

typos or errors (notably Tom Bullock). Their efforts have greatly improved the readability of these

notes.

c b a n

8 PHY472: Advanced Quantum Mechanics

If two vectors have an inner product equal to zero, then these vectors are called orthogonal.

This is the definition of orthogonality. When these vectors are also unit vectors, they are called

orthonormal. A set of vectors

φ 1

φ 2

φ N

are linearly independent if

j

a j

φ j

implies that all a j

= 0. The maximum number of linearly independent vectors in V is the di-

mension of V. Orthonormal vectors form a complete orthonormal basis for V if any vector can be

written as

ψ

N ∑

k= 1

c k

φ k

and 〈 φ j

| φ k

〉 = δ jk

. We can take the inner product of

ψ

with any of the basis vectors

φ j

to obtain

〈 φ j

| ψ 〉 =

N ∑

k= 1

c k

〈 φ j

| φ k

N ∑

k= 1

c k

δ jk

= c j

Substitute this back into the expansion of

ψ

, and we find

ψ

N ∑

k= 1

φ k

〈 φ k

| ψ 〉. (1.5)

Therefore

k

φ k

φ k

must act like the identity. In fact, this gives us an important clue that

operators of states must take the general form of sums over objects like

φ

χ

1.2 Operators in Hilbert space

The objects

ψ

are vectors in a Hilbert space. We can imagine applying rotations of the vectors,

rescaling, permutations of vectors in a basis, and so on. These are described mathematically as

operators, and we denote them by capital letters A, B, C, etc. In general we write

A

φ

ψ

for some

φ

ψ

∈ V. It is important to remember that operators act on all the vectors in Hilbert

space. Let {

φ j

j

be an orthonormal basis. We can calculate the inner product between the

vectors

φ j

and A

φ k

〈 φ j

A

φ k

= 〈 φ j

|A| φ k

〉 ≡ A

jk

The two indices indicate that operators are matrices.

As an example, consider two vectors, written as two-dimensional column vectors

φ 1

φ 2

and suppose that

A =

Section 1: Linear Vector Spaces and Hilbert Space 9

We calculate

A 11 =

φ 1

A

φ 1

Similarly, we can calculate that A 22 = 3, and A 12 = A 21 = 0 (check this). We therefore have that

A

φ 1

φ 1

and A

φ 2

φ 2

Complex numbers a have complex conjugates a

∗ and vectors

ψ

have dual vectors

φ

. Is

there an equivalent for operators? The answer is yes, and it is called the adjoint, or Hermitian

conjugate, and is denoted by a dagger †. The natural way to define it is according to the rule

ψ

A

φ

∗

=

φ

A

†

ψ

for any

φ

and

ψ

. In matrix notation, and given an orthonormal basis {

φ j

j

, this becomes

φ j

A

φ k

∗

= A

∗

jk

φ k

A

†

φ j

= A

†

k j

So the matrix representation of the adjoint A

† is the transpose and the complex conjugate of the

matrix A, as given by (A

†

) jk

= A

∗

k j

. The adjoint has the following properties:

1. (cA)

† = c

∗ A

† ,

2. (AB)

†

= B

†

A

†

,

φ

†

=

φ

Note the order of the operators in 2: AB is generally not the same as BA. The difference between

the two is called the commutator, denoted by

[A, B] = AB − BA. (1.13)

For example, we can choose

A =

and B =

which leads to

[A, B] =

Many, but not all, operators have an inverse. Let A

φ

ψ

and B

ψ

φ

. Then we have

BA

φ

φ

and AB

ψ

ψ

If Eq. (1.16) holds true for all

φ

and

ψ

, then B is the inverse of A, and we write B = A

− 1

. An

operator that has an inverse is called invertible. Another important property that an operator

may possess is positivity. An operator is positive if

φ

A

φ

≥ 0 for all

φ

We also write this as A ≥ 0.

Section 1: Linear Vector Spaces and Hilbert Space 11

Note that this is independent of the basis {

a (^) j

}. As a consequence, for any orthonormal basis

φ j

} we have

I =

j

φ j

φ j

This is the completeness relation, and we will use this many times in our calculations.

Lemma: If two non-degenerate operators commute ([A, B] = 0), then they have a common set of

eigenvectors.

Proof: Let A =

k

a k

|a k

〉 〈a k

| and B =

jk

B

jk

a (^) j

〈a k

|. We can choose this without loss of gener-

ality: we write both operators in the eigenbasis of A. Furthermore, [A, B] = 0 implies that

AB = BA.

AB =

klm

a k

B

lm

|a k

〉 〈a k

|a l

〉 〈a m

lm

a l

B

lm

|a l

〉 〈a m

BA =

klm

a k

B

lm

|a l

〉 〈a m

|a k

〉 〈a k

lm

a m

B

lm

|a l

〉 〈a m

Therefore

[A, B] =

lm

(a l

− a m

)B

lm

|a l

〉 〈a m

If a l

= a m

for l 6 = m, then B lm

= 0, and B lm

∝ δ lm

. Therefore {

a j

} is an eigenbasis for B. ‰

The proof of the converse is left as an exercise. It turns out that this is also true when A and/or

B are degenerate.

1.3 Hermitian and unitary operators

Next, we will consider two special types of operators, namely Hermitian and unitary operators.

An operator A is Hermitian if and only if A

†

= A.

Lemma: An operator is Hermitian if and only if it has real eigenvalues: A

†

= A ⇔ a j

∈ R.

Proof: The eigenvalue equation of A implies that

A

a j

= a j

a j

a j

A

†

= a

∗

j

a j

which means that

a j

A

a j

= a j

and

a j

A

†

a j

= a

∗

j

. It is now straightforward to show

that A = A

† implies a j

= a

∗

j

, or a j

∈ R. Conversely, a j

∈ R implies a j

= a

∗

j

, and

a (^) j

A

a (^) j

a (^) j

A

†

a (^) j

Let

ψ

k

c k

|a k

〉. Then

ψ

A

ψ

j

|c (^) j|

2

a (^) j

A

a (^) j

j

|c (^) j|

2

a (^) j

A

†

a (^) j

j

|c (^) j|

2

a (^) j

A

†

a (^) j

ψ

A

†

ψ

for all

ψ

, and therefore A = A

†

. ‰

12 PHY472: Advanced Quantum Mechanics

Next, we construct the exponent of an operator A according to U = exp(icA). We have included

the complex number c for completeness. At first sight, you may wonder what it means to take the

exponent of an operator. Recall, however, that the exponent has a power expansion:

U = exp(icA) =

∞ ∑

n= 0

(ic)

n

n!

A

n

. (1.30)

The n

th power of an operator is straightforward: just multiply A n times with itself. The expres-

sion in Eq. (1.30) is then well defined, and the exponent is taken as an abbreviation of the power

expansion. In general, we can construct any function of operators, as long as we can define the

function in terms of a power expansion:

f (A) =

∞ ∑

n= 0

fn A

n

. (1.31)

This can also be extended to functions of multiple operators, but now we have to be very careful

in the case where these operators do not commute. Familiar rules for combining normal functions

no longer apply (see exercise 4b).

We can construct the adjoint of the operator U according to

U

†

=

∞ ∑

n= 0

(ic)

n

n!

A

n

†

∞ ∑

n= 0

(−ic

∗ )

n

n!

A

†n

= exp(−ic

∗

A

†

). (1.32)

In the special case where A = A

† and c is real, we calculate

UU

†

= exp(icA) exp(−ic

∗

A

†

) = exp(icA) exp(−icA) = exp[ic(A − A)] = I , (1.33)

since A commutes with itself. Similarly, U

† U = I. Therefore, U

† = U

− 1 , and an operator with this

property is called unitary. Each unitary operator can be generated by a Hermitian (self-adjoint)

operator A and a real number c. A is called the generator of U. We often write U = U A

(c). Unitary

operators are basis transformations.

1.4 Projection operators and tensor products

We can combine two linear vector spaces U and V into a new linear vector space W = U ⊕ V. The

symbol ⊕ is called the direct sum. The dimension of W is the sum of the dimensions of U and V :

dim W = dim U + dim V. (1.34)

A vector in W can be written as

W

ψ

U

φ

V

where

ψ

U

and

φ

V

are typically not normalized (i.e., they are not unit vectors). The spaces U

and V are so-called subspaces of W.

As an example, consider the three-dimensional Euclidean space spanned by the Cartesian

axes x, y, and z. The x y-plane is a two-dimensional subspace of the full space, and the z-axis is

a one-dimensional subspace. Any three-dimensional form can be projected onto the x y-plane by

setting the z component to zero. Similarly, we can project onto the z-axis by setting the x and y

coordinates to zero. A projector is therefore associated with a subspace. It acts on a vector in the

full space, and forces all components to zero, except those of the subspace it projects onto.

14 PHY472: Advanced Quantum Mechanics

2. I ⊗ I = I,

3. (A

1

+ A

2

) ⊗ B = A

1

⊗ B + A

2

⊗ B,

4. aA ⊗ bB = (ab)A ⊗ B,

5. (A ⊗ B)

− 1 = A

− 1 ⊗ B

− 1 ,

6. (A ⊗ B)

† = A

† ⊗ B

† .

Note that the last rule preserves the order of the operators. In other words, operators always act

on their own space. Often, it is understood implicitly which operator acts on which subspace, and

we will write A ⊗I = A and I⊗ B = B. Alternatively, we can add subscripts to the operator, e.g., A U

and B V

As a practical example, consider two two-dimensional operators

A =

A 11 A 12

A

21

A

22

and B =

B 11 B 12

B

21

B

22

with respect to some orthonormal bases {|a 1

〉 , |a 2

〉} and {|b 1

〉 , |b 2

〉} for A and B, respectively (not

necessarily eigenbases). The question is now: what is the matrix representation of A ⊗ B? Since

the dimension of the new vector space is the product of the dimensions of the two vector spaces,

we have dim W = 2 · 2 = 4. A natural basis for A ⊗ B is then given by {

a j

, b k

jk

, with j, k = 1 , 2, or

|a 1

〉 |b 1

〉 , |a 1

〉 |b 2

〉 , |a 2

〉 |b 1

〉 , |a 2

〉 |b 2

We can construct the matrix representation of A ⊗B by applying this operator to the basis vectors

in Eq. (1.48), using

A

a (^) j

= A 1 j |a 1 〉 + A 2 j |a 2 〉^ and B |ak〉 = B 1 k |b 1 〉 + B 2 k |b 2 〉^ , (1.49)

which leads to

A ⊗ B |a 1

, b 1

〉 = (A

11

|a 1

〉 + A

21

|a 2

〉)(B

11

|b 1

〉 + B

21

|b 2

A ⊗ B |a 1

, b 2

〉 = (A

11

|a 1

〉 + A

21

|a 2

〉)(B

12

|b 1

〉 + B

22

|b 2

A ⊗ B |a 2

, b 1

〉 = (A

12

|a 1

〉 + A

22

|a 2

〉)(B

11

|b 1

〉 + B

21

|b 2

A ⊗ B |a 2

, b 2

〉 = (A

12

|a 1

〉 + A

22

|a 2

〉)(B

12

|b 1

〉 + B

22

|b 2

Looking at the first line of Eq. (1.50), the basis vector |a 1

, b 1

〉 gets mapped to all basis vectors:

A ⊗ B |a 1 , b 1 〉 = A 11 B 11 |a 1 , b 1 〉 + A 11 B 21 |a 1 , b 2 〉 + A 21 B 11 |a 2 , b 1 〉 + A 21 B 21 |a 2 , b 2 〉. (1.51)

Combining this into matrix form leads to

A ⊗ B =

A

11

B

11

A

11

B

12

A

12

B

11

A

12

B

12

A

11

B

21

A

11

B

22

A

12

B

21

A

12

B

22

A 21 B 11 A 21 B 12 A 22 B 11 A 22 B 12

A

21

B

21

A

21

B

22

A

22

B

21

A

22

B

22

A

11

B A

12

B

A 21 B A 22 B

Recall that this is dependent on the basis that we have chosen. In particular, A ⊗ B may be

diagonalized in some other basis.

Section 1: Linear Vector Spaces and Hilbert Space 15

1.5 The trace and determinant of an operator

There are two special functions of operators that play a key role in the theory of linear vector

spaces. They are the trace and the determinant of an operator, denoted by Tr(A) and det(A),

respectively. While the trace and determinant are most conveniently evaluated in matrix repre-

sentation, they are independent of the chosen basis.

When we defined the norm of an operator, we introduced the trace. It is evaluated by adding

the diagonal elements of the matrix representation of the operator:

Tr(A) =

j

φ j

A

φ j

where {

φ j

j

is any orthonormal basis. This independence means that the trace is an invariant

property of the operator. Moreover, the trace has the following important properties:

1. If A = A

†

, then Tr(A) is real,

2. Tr(aA) = aTr(A),
3. Tr(A + B) = Tr(A) + Tr(B),
4. Tr(AB) = Tr(BA) (the "cyclic property").

The first property follows immediately when we evaluate the trace in the diagonal basis, where it

becomes a sum over real eigenvalues. The second and third properties convey the linearity of the

trace. The fourth property is extremely useful, and can be shown as follows:

Tr(AB) =

j

φ j

AB

φ j

jk

φ j

A

ψ k

ψ k

B

φ j

jk

ψ k

B

φ j

φ j

A

ψ k

k

ψ k

BA

ψ k

= Tr(BA). (1.54)

This derivation also demonstrates the usefulness of inserting a resolution of the identity in strate-

gic places. In the cyclic property, the operators A and B may be products of two operators, which

then leads to

Tr(ABC) = Tr(BC A) = Tr(C AB). (1.55)

Any cyclic (even) permutation of operators under a trace gives rise to the same value of the trace

as the original operator ordering.

Finally, we construct the partial trace of an operator that lives on a tensor product space.

Suppose that A ⊗ B is an operator in the Hilbert space H 1 ⊗ H 2. We can trace out Hilbert space

H

1

, denoted by Tr 1

Tr 1

(A ⊗ B) = Tr(A)B , or equivalently Tr 1

(A

1

B

2

) = Tr(A 1

)B

2

Taking the partial trace has the effect of removing the entire Hilbert space H 1

from the de-

scription. It reduces the total vector space. The partial trace always carries an index, which

determines which space is traced over.

The determinant of a 2 × 2 matrix is given by

det(A) = det

A

11

A

12

A

21

A

22

= A

11

A

22

− A

12

A

21

Section 1: Linear Vector Spaces and Hilbert Space 17

(c) Let U be a transformation matrix that maps one complete orthonormal basis to an-

other. Show that U is unitary.

(d) How many real parameters completely determine a d × d unitary matrix?

3. Properties of the trace and the determinant:

(a) Calculate the trace and the determinant of the matrices A and B in exercise 1c.

(b) Show that the expectation value of A can be written as Tr(

ψ

ψ

A).

(c) Prove that the trace is independent of the basis.

4. Commutator identities.

(a) Let F(t) = e

At e

Bt

. Calculate dF/dt and use [e

At , B] = (e

At Be

−At −B)e

At to simplify your

result.

(b) Let G(t) = e

At+Bt+ f (t)H

. Show by calculating dG/dt, and setting dF/dt = dG/dt at t = 1,

that the following operator identity

e

A

e

B

= e

A+B+

1

2

[A,B]

, (1.63)

holds if A and B both commute with [A, B]. Hint: use the Hadamard lemma

e

At

Be

−At

= B +

t

[A, B] +

t

2

[A, [A, B]] +... (1.64)

(c) Show that the commutator of two Hermitian operators is anti-Hermitian (A

† = −A).

(d) Prove the commutator analog of the Jacobi identity

[A, [B, C]] + [B, [C, A]] + [C, [A, B]] = 0. (1.65)

18 PHY472: Advanced Quantum Mechanics

2 The Postulates of Quantum Mechanics

The entire structure of quantum mechanics (including its relativistic extension) can be formulated

in terms of states and operations in Hilbert space. We need rules that map the physical quantities

such as states, observables, and measurements to the mathematical structure of vector spaces,

vectors and operators. There are several ways in which this can be done, and here we summarize

these rules in terms of five postulates.

Postulate 1: A physical system is described by a Hilbert space H , and the state of the system is

represented by a ray with norm 1 in H.

There are a number of important aspects to this postulate. First, the fact that states are rays,

rather than vectors means that an overall phase e

i ϕ of the state does not have any physically

observable consequences, and e

i ϕ

ψ

represents the same state as

ψ

. Second, the state contains

all information about the system. In particular, there are no hidden variables in this standard

formulation of quantum mechanics. Finally, the dimension of H may be infinite, which is the

case, for example, when H is the space of square-integrable functions.

As an example of this postulate, consider a two-level quantum system (a qubit). This system

can be described by two orthonormal states | 0 〉 and | 1 〉. Due to linearity of Hilbert space, the

superposition α | 0 〉 + β | 1 〉 is again a state of the system if it has norm 1, or

( α

∗ 〈 0 | + β

∗ 〈 1 |)( α | 0 〉 + β | 1 〉) = 1 or | α |

2

• | β |

2

= 1. (2.1)

This is called the superposition principle: any normalised superposition of valid quantum states

is again a valid quantum state. It is a direct consequence of the linearity of the vector space, and

as we shall see later, this principle has some bizarre consequences that have been corroborated

in many experiments.

Postulate 2: Every physical observable A corresponds to a self-adjoint (Hermitian

1

) operator

A

whose eigenvectors form a complete basis.

We use a hat to distinguish between the observable and the operator, but usually this distinction

is not necessary. In these notes, we will use hats only when there is a danger of confusion.

As an example, take the operator X :

X | 0 〉 = | 1 〉 and X | 1 〉 = | 0 〉. (2.2)

This operator can be interpreted as a bit flip of a qubit. In matrix notation the state vectors can

be written as

and | 1 〉 =

which means that X is written as

X =

with eigenvalues ±1. The eigenstates of X are

p

These states form an orthonormal basis.

1 In Hilbert spaces of infinite dimensionality, there are subtle differences between self-adjoint and Hermitian

operators. We ignore these subtleties here, because we will be mostly dealing with finite-dimensional spaces.

20 PHY472: Advanced Quantum Mechanics

We can write the state of a system at time t as

ψ (t)

, and at some time t 0 < t as

ψ (t 0 )

. The

fourth postulate tells us that there is a unitary operator U(t, t 0

) that transforms the state at time

t 0

to the state at time t:

ψ (t)

= U(t, t 0

ψ (t 0

Since the evolution from time t to t is denoted by U(t, t) and must be equal to the identity, we

deduce that U depends only on time differences: U(t, t 0

) = U(t − t 0

), and U(0) = I.

As an example, let U(t) be generated by a Hermitian operator A according to

U(t) = exp

i

ħ

At

The argument of the exponential must be dimensionless, so A must be proportional to ħ times

an angular frequency (in other words, an energy). Suppose that

ψ (t)

is the state of a qubit, and

that A = ħ ω X. If

ψ (0)

= | 0 〉 we want to calculate the state of the system at time t. We can write

ψ (t)

= U(t)

ψ (0)

= exp (−i ω tX (^) ) | 0 〉 =

∞ ∑

n= 0

(−i ω t)

n

n!

X

n

. (2.16)

Observe that X

2 = I, so we can separate the power series into even and odd values of n:

ψ (t)

∞ ∑

n= 0

(−i ω t)

2 n

(2n)!

∞ ∑

n= 0

(−i ω t)

2 n+ 1

(2n + 1)!

X | 0 〉 = cos( ω t) | 0 〉 − i sin( ω t) | 1 〉^. (2.17)

In other words, the state oscillates between | 0 〉 and | 1 〉.

The fourth postulate also leads to the Schrödinger equation. Let’s take the infinitesimal form

of Eq. (2.14):

ψ (t + dt)

= U(dt)

ψ (t)

We require that U(dt) is generated by some Hermitian operator H:

U(dt) = exp

i

ħ

H dt

H must have the dimensions of energy, so we identify it with the energy operator, or the Hamil-

tonian. We can now take a Taylor expansion of

ψ (t + dt)

to first order in dt:

ψ (t + dt)

ψ (t)

• dt

d

dt

ψ (t)

and we expand the unitary operator to first order in dt as well:

U(dt) = 1 −

i

ħ

Hdt +... (2.21)

We combine this into

ψ (t)

• dt

d

dt

ψ (t)

i

ħ

Hdt

ψ (t)

which can be recast into the Schrödinger equation:

iħ

d

dt

ψ (t)

= H

ψ (t)

Therefore, the Schrödinger equation follows directly from the postulates!

Section 2: The Postulates of Quantum Mechanics 21

Figure 1: Schrödingers Cat.

Postulate 5: If a measurement of an observable A yields an eigenvalue a j

, then immediately

after the measurement, the system is in the eigenstate

a (^) j

corresponding to the eigenvalue.

This is the infamous projection postulate, so named because a measurement “projects” the system

to the eigenstate corresponding to the measured value. This postulate has as observable conse-

quence that a second measurement immediately after the first will also find the outcome a j

. Each

measurement outcome a (^) j corresponds to a projection operator P (^) j on the subspace spanned by the

eigenvector(s) belonging to a j

. A (perfect) measurement can be described by applying a projector

to the state, and renormalize:

ψ

P (^) j

ψ

∥P

j

ψ

This also works for degenerate eigenvalues.

We have established earlier that the expectation value of A can be written as a trace:

〈A〉 = Tr(

ψ

ψ

A). (2.25)

Now instead of the full operator A, we calculate the trace of P j

a j

a j

〈P

j

〉 = Tr(

ψ

ψ

P

j

) = Tr(

ψ

〈 ψ |a j

a j

) = |〈a j

| ψ 〉|

2

= p(a j

So we can calculate the probability of a measurement outcome by taking the expectation value of

the projection operator that corresponds to the eigenstate of the measurement outcome. This is

one of the basic calculations in quantum mechanics that you should be able to do.

The projection postulate is somewhat problematic for the interpretation of quantum mechan-

ics, because it leads to the so-called measurement problem: Why does a measurement induce a

non-unitary evolution of the system? After all, the measurement apparatus can also be described

quantum mechanically

2

and then the system plus the measurement apparatus evolves unitarily.

But then we must invoke a new device that measures the combined system and measurement

apparatus. However, this in turn can be described quantum mechanically, and so on.

On the other hand, we do see definite measurement outcomes when we do experiments, so at

some level the projection postulate is necessary, and somewhere there must be a “collapse of the

wave function”. Schrödinger already struggled with this question, and came up with his famous

2 This is something most people require from a fundamental theory: quantum mechanics should not just break

down for macroscopic objects. Indeed, experimental evidence of macroscopic superpositions has been found in the

form of “cat states”.