Quantum Mechanics 1 Lecture 3: Formalism and Postulates of QM

Quantum Mechanics 1 Lecture 3:

Formalism and Postulates of QM

Massimiliano Grazzini

University of Zurich

Outline

Hilbert spaces

Observables and operators

Correspondence principle

Ehrenfest theorem Momentum operator

Wrap Up: The postulates of Quantum Mechanics General form of the uncertainty principle

Time evolution

Wave packets

Observables and Operators

Let us consider a physical observable that characterises a quantum system _𝒜 The values that this quantity can assume are elements of the spectrum of this physical quantity

In classical mechanics physical quantities take in general continuous values In quantum mechanics there can also be quantities with a spectrum of

discrete values: for simplicity we will assume this is the case for the moment We call ( ) the discrete values of and we call the wave

function of the system in the state in which the quantity takes the value ^a

^k = 0,1,2.... 𝒜 ϕ

𝒜 a

We assume

∫ | ϕ

|

dx = 1

If our system is an arbitrary state characterised by a wave function the measurement of the quantity will give one of the values _{𝒜 a}

^ψ

3

According to the superposition principle, we can then say that a generic wave function of the system will be represented by

ψ = ∑

k

c

ϕ

Such expansion allows us to determine the probability to obtain (through a measurement) a value of the quantity _a

_𝒜

Such probability must be positive definite, must be unity if , and must

vanish if _c

_{= 0} ^{ψ = ϕ}

c

∈ ℂ

the probability should be _{| c}

_|

The sum of these probabilities should be unity, so we have the constraint

∑

| c

|

= ∫ | ψ |

dx

If we multiply _{ψ * = ∑} by and we integrate we obtain

k

c*

ϕ*

ψ

∫ | ψ |

dx = ∑

k

c*

∫ ϕ*

ψ dx

4

Comparing with the previous expression we find

5

c

= ∫ ψ ϕ*

dx

and if we plug in this equation the expansion _{ψ = ∑} we see that

l

c

ϕ

∫ ϕ

ϕ*

dx = δ

So, we arrive at a structure of vector space with scalar product (recall MMP1 !) of which the functions form an orthonormal basis _ϕ

We now want to introduce the concept of mean (expectation) value of the quantity _𝒜

⟨𝒜⟩ ≡ ∑

k

a

| c

|

We can define it as a weighted average of the values where each of them is multiplied by the probability _{| c}

_|

^a

(1)

6

We now would like to express _⟨𝒜⟩ directly in terms of _ψ

Since _⟨𝒜⟩ is bilinear in and such expression must be bilinear in and _c

_c*

_{ψ ψ *} We introduce an operator on the functions such that _{A ψ}

⟨𝒜⟩ = ∫ ψ *(Aψ) dx

By using _{⟨𝒜⟩ ≡ ∑} and we obtain

k

a

c

c*

c*

= ∫ ψ* ϕ

dx

⟨𝒜⟩ = ∫ ψ * ∑

k

c

a

ϕ

dx

Comparing with Eq. we see that the action of the operator on is _{(2) A ψ}

(2)

Aψ = ∑

k

a

c

ϕ

Using Eq. for we obtain _{(1) c}

(Aψ)(x) = ∫ ∑

k

a

ϕ*

(x′ )ϕ

(x)ψ (x′ )dx′ ≡ ∫ K

(x, x′ )ψ (x′ )dx′

7

(Aψ)(x) = ∫ ∑

k

a

ϕ*

(x′ )ϕ

(x)ψ (x′ )dx′ ≡ ∫ K

(x, x′ )ψ (x′ )dx′

We thus conclude that the operator acts as an integral linear operator on _{A ψ} To each physical quantity in quantum mechanics

we associate an appropriate linear operator From _{Aψ = ∑} we see that if we have and

k

a

c

ϕ

ψ = ϕ

c

= δ

Aϕ

= a

ϕ

So, the action of on is simply to multiply it by the corresponding and the eigenfunctions of the operator are just the solutions of the equation ^{A ϕ}

_A ^a

Aψ = a ψ

The fact that the values that a physical quantity can take are real poses some limitations on the properties of the corresponding operator ^𝒜 _A

⟨𝒜⟩ = ∫ ψ *(Aψ) dx = ⟨𝒜⟩* = ∫ (Aψ)*ψdx

On the other hand the adjoint operator is defined as _{∫ ψ *(A χ) dx = ∫ (A}

_{ψ)* χdx}

8

We thus conclude that the operator corresponding to a physical quantity must

be self-adjoint ^A _{A = A}

We now consider the case in which we have two physical quantities and that

can be simultaneously measured ^{𝒜 ℬ}

In this case the two operators must have common eigenfunctions

The product of the corresponding operators and can be defined as _{A B}

(AB)ψ ≡ A(B ψ)

If we apply the product on one of the common eigenfunctions we have _{AB ϕ}

(AB)ϕ

= A(Bϕ

) = A(b

ϕ

) = b

(Aϕ

) = b

a

ϕ

On the other hand

(BA)ϕ

= B(Aϕ

) = B(a

ϕ

) = a

(Bϕ

) = a

b

ϕ

We conclude that on each eigenfunction we have but since each wave-function can be expanded in this basis this must hold in general ^ϕ

^{[A, B] ≡ AB − BA = 0}

If two physical quantities can be simultaneously

measured their operators commute

Hilbert spaces

9

The superposition principle and the idea that the wave-function describes the state of a system require the mathematical foundations of QM to be laid

down by using linear spaces of functions with scalar products, which allow us to define normalisation and transition rates: these are the Hilbert spaces

Recall MMP1: Hilbert space: “vector space with scalar product which is complete with respect to the metric induced by the norm”

We consider a vector space on a field ( _{ℋ 𝕂 𝕂 = ℝ} or ) _ℂ

A scalar product _{⟨ ⋅ | ⋅ ⟩} is a mapping from _{ℋ × ℋ → 𝕂} such that ( _{| x⟩, | y⟩, | z⟩ ∈ ℋ} ) (i) _{⟨x | α y + βz⟩ = α⟨x | y⟩ + β⟨x | z⟩}

(ii) _{⟨x | y⟩ = ⟨y | x⟩*}

(iii) _{⟨x | x⟩ ≥ 0} and _{⟨x | x⟩ = 0 ⇔ x = 0}

α, β ∈ 𝕂

Dirac “ket” notation

Hilbert spaces

10

Given a vector space with scalar product, a norm is defined through _ℋ

∥x∥ = ⟨x | x⟩

A vector space with scalar product is a Hilbert space if it is complete with respect to the metric induced by the norm ^ℋ

This means that every Cauchy sequence is such that _ϕ

_lim

n→∞

ϕ

∈ ℋ

In general _{dim ℋ = ∞}

In Quantum Mechanics we will limit ourselves to consider the case in which the Hilbert space is separable: this means that a complete

numerable orthonormal basis exists _{ϕ

_{, n} = 1,2...}

Examples:

- _{ℋ = ℂ}

: prototype of complex vector space with finite dimension

- _{ℋ = L}

_{[a, b]} : space of the square-integrable functions in the interval _{[a, b]}

11

Momentum Operator

In position space the probability to find a particle at the position in the

element is given by _dx ^x

| ψ (x, t) |

dx

Correspondingly, let us define the probability of finding the particle with momentum between and _p ^w( _p ^p, ₊ ^t)dp _dp

We require

∫ dpw( p, t) = 1

By writing _{ψ (x, t) =}

∫ dp

2π ℏ ψ ˜ ( p, t)e

we find

∫ dx | ψ (x, t) |

= ∫ dx ∫ dp

2π ℏ dp′

2πℏ e

ψ ˜ ( p, t) ˜ ψ ( p′ , t)*

= ∫ dp

2π ℏ | ψ ˜ ( p, t) |

Use _{∫ dx exp{i / ℏ( p − p′ )x} = 2π ℏδ( p − p′ )}

12

This result suggests the following definition for the probability density in momentum space

w( p, t) = 1 2π ℏ | ψ ˜ ( p, t) |

This is consistent with the idea that for a plane wave with momentum the Fourier transform _{ψ ˜ ( p, t)} differs from zero only for _{p = p}

^p

We can now compute the expectation value _⟨p⟩ of the momentum as

⟨p⟩ = ∫ dp

2πℏ ψ ˜ *( p, t) p ψ ˜ ( p, t) = ∫ dxdx′ ∫ dp

2πℏ e

ψ*(x′ , t) p e

ψ (x, t)

If we write _pe

_{ψ (x, t) = iℏ (} ^∂

∂x e

) ψ (x, t) = − iℏe

∂ ψ (x, t)

∂x

integrating by parts

⟨p⟩ = ∫ dp p w( p, t)

We find

˜

ψ ( p, t) = ∫ dx ψ (x, t)e

˜

ψ *( p, t) = ∫ dx′ ψ *(x′ , t)e

use

13

p = − iℏ ∂

∂x

We thus conclude that the operator corresponding to the momentum is _p

This result is easily generalised to the tridimensional case in which the wave function is a function of the position vector _x

In this case we have

p = − iℏ∇

⟨ p⟩ = ∫ dxdx′ ∫ dp

2πℏ e

ψ *(x′ , t)(− iℏ)e

∂ψ (x, t)

∂x = ∫ dxdx′ δ(x − x′ )ψ *(x′ , t)(− iℏ) ∂ψ (x, t)

∂x ψ (x, t)

= ∫ dx ψ *(x, t) (− iℏ) ∂

∂x

Continuous spectrum

We have seen that in the case of an observable with a discrete spectrum of values corresponding to the eigenfunctions a general wave function can be expanded as

a

ϕ

𝒜

ψ = ∑

k

c

ϕ

c

∈ ℂ

_{Aψ = ∑} and

k

a

c

ϕ

Aϕ

= a

ϕ

And the corresponding operator is such that _A

The momentum operator is an example of operator with continuous spectrum Limiting ourselves to the one-dimensional case the eigenvalue equation reads

pϕ

(x) = − iℏ ∂ ∂x ϕ

(x) = pϕ

(x)

And the solution is

ϕ

(x) = (2π ℏ)

e

The functions form a complete orthonormal system as the introduced

for the discrete spectrum ^ϕ

^{(x) ϕ}

The only difference is that the sum over is replaced by an integral over and that the condition is replaced by

k p

∫ ϕ

ϕ*

dx = δ

∫ ϕ*

(x)ϕ

(x)dx = δ( p − p′ )

and the completeness relation reads

∫ ϕ*

(x′ )ϕ

(x)dp = δ(x − x′ )

The expansion of an arbitrary wave function in momentum eigenfunctions is nothing but a Fourier expansion and reads

ψ (x) = 1

2π ℏ ∫ dpc

ϕ

(x) = 1

2π ℏ ∫ dpc

e

or, by using _{k = p / ℏ}

ψ (x) = 1

2π ∫ dkc(k)e

16

The Correspondence principle

We can thus summarise what we have learned so far and connect it with the mathematical structure of a Hilbert space

The wave functions _{ψ (x, t)} become vectors in Hilbert spaces _{| ψ⟩}

The observables become linear hermitian operators on such Hilbert spaces _𝒜 The assignment of the classical measured variables to the quantum

mechanical observables is governed by the correspondence principle:

Position _{x x}

Momentum p − iℏ∇

Energy _{E iℏ∂ / ∂t}

To what extent can we assign quantum mechanical relations to classical

relations on the basis of this correspondence ?

17

For example, if we take the classical energy-momentum relation for a massive particle in a potential _V(x)

E = p

2m + V

Does it mean that ? _{iℏ(∂ / ∂t) = − (ℏ}

_{/2m) ∇}

_{+ V}

This relation cannot be valid at the operator level, but when applied on the wave function it gives exactly the Schrödinger equation

iℏ(∂ψ / ∂t) = Hψ H = − (ℏ

/2m) ∇

+ V(x)

where

is the Hamilton operator

18

Linear operators and commutators

In Quantum Mechanics the relevant operators are linear

| ψ

⟩, | ψ

⟩ ∈ H c

, c

∈ ℂ A ( c

| ψ

⟩ + c

| ψ

⟩ ) = c

A | ψ

⟩ + c

A | ψ

⟩

Examples: _x

_, ^∂ as multiplier….

∂x

, ∇

, ∂

∂t , f (x, t)

We have seen that if two physical quantities can be simultaneously measured the corresponding operators must commute

it is clear that _{[ x}

_{, x}

_{] = 0} Since _(x

_x

_{− x}

_x

_{)ψ = 0}

Let us consider the commutator

[ x

, ∂

∂x

]

The opposite is also true

19

Linear operators and commutators

[ x

, ∂

∂x

] = − δ

We thus conclude that We have

[ x

, ∂

∂x

] ψ =

( x

∂

∂x

− ∂ ∂x

x

) ψ = x

∂ψ

∂x

− δ

ψ − x

∂ψ

∂x

= − δ

ψ

Since _p

_{= − iℏ∇}

we see that _{[ x}

_{, p}

_{] = iℏδ}

This result shows that spatial and

momentum components do not commute

This is consistent with the uncertainty principle! it is not possible to exactly

measure position and momentum at the same time

Measurement process

We consider again a physical observable that characterises the quantum system and is the corresponding linear operator _A ^𝒜

Let ( ) be the discrete values of and the wave function of the system in the state in which the quantity takes the value ^a

^k = 0,1,2.... 𝒜 ϕ

𝒜 a

ψ = ∑

k

c

ϕ

c

∈ ℂ

A generic state can be expanded as _ψ

And the coefficients give the probability that a measurement of returns the value ^{| c}

^|

2

𝒜

a

After the measurement returns one of the values the system will be in the

state _ϕ

^a

| ψ⟩ measurement of with result

| ϕ

⟩

This is known as “wave function collapse” (or von Neumann projection)

21

General form of the uncertainty principle

We now want to give the general form of the uncertainty principle

We start from a general property of a vector space with scalar product _ℋ Cauchy-Schwarz inequality (recall MMP1 !)

∀ | ϕ⟩, | ψ⟩ ∈ ℋ | ⟨ϕ | ψ⟩ | ≤ ∥ϕ∥∥ψ ∥

We then consider two Hermitian operators and and an arbitrary state _A

_A

_ψ We define the operators obtained by subtracting from and their expectation value on the state

̂ A

= A

− ⟨A

⟩ = A

− ⟨ψ | A

| ψ⟩

A

A

ψ

We use the Cauchy-Schwarz inequality with _A

^̂ _ψ and _A

^̂ _ψ

| ⟨ ̂ A

ψ | A

̂ ψ⟩ | ≤ ∥ ̂ A

ψ ∥∥ ̂ A

ψ ∥

or | ⟨ ̂ A

ψ | A

̂ ψ⟩ |

≤ ⟨ ̂ A

ψ | A

̂ ψ⟩⟨ ̂ A

ψ | A ̂

ψ⟩

22

Using the fact that and are Hermitian we have _A

_A

| ⟨ψ | A ̂

A ̂

ψ⟩ |

≤ ⟨ψ | A ̂

| ψ⟩⟨ψ | A

̂ | ψ⟩

Now, even if the operators and are Hermitian their product is not: indeed _A ^̂

_A

^̂

( A

̂ A

̂ )

= A

̂ A

̂ = A ̂

A ̂

≠ ̂ A

A ̂

We decompose the product _A

^̂ _A

^̂ into an Hermitian and anti-Hermitian part

̂ A

̂ A

= 1 2 ( ̂ A

̂ A

− ̂ A

̂ A

) + 1 2 ( ̂ A

̂ A

+ ̂ A

̂ A

) = 1 2 [ ̂ A

, ̂ A

] + 1 2 { ̂ A

, ̂ A

}

Indeed we have

[ A

̂ , A

̂ ]

= − [ A

̂ , A

̂ ] { A

̂ , A

̂ }

= { A ̂

, A ̂

}

anti-Hermitian Hermitian

Hermitian and anti-Hermitian operators have real and purely imaginary expectation values

⟨ψ | Aψ⟩* = ⟨Aψ | ψ⟩ = ⟨ψ | Aψ⟩ _A Hermitian

⟨ψ | Aψ⟩* = ⟨Aψ | ψ⟩ = − ⟨ψ | Aψ⟩ _A anti-Hermitian (3)

use _⟨ψ

_{| A}

^̂ _A

^̂ _{| ψ}

_{⟩ = ⟨ ̂ A}

_ψ

_{| A}

^̂ _{| ψ}

_{⟩ = ⟨ ̂ A}

_A

^̂ _ψ

_{| ψ}

_⟩

23

This means that when computing _{| ⟨ψ | A}

^̂ _A

^̂ _{ψ⟩ |}

we can write

| ⟨ψ | A ̂

A

̂ ψ⟩ |

= 1

4 | ⟨ψ | { A

̂ , A ̂

}ψ⟩ |

+ 1

4 | ⟨ψ | [ A

̂ , A

̂ ]ψ⟩ |

Since expectation values are ordinary numbers we have _{[ A}

^̂ _{, A}

^̂ _{] = [A}

_{, A}

_] And from the previous result we conclude that

| ⟨ψ | A

̂ A ̂

ψ⟩ |

≥ 1

4 | ⟨ψ | [A

, A

]ψ⟩ |

(4)

The uncertainty _{Δ A} is defined as the root-mean-square deviation

(Δ A)

≡ ⟨ψ | (A − ⟨A⟩)

| ψ⟩

Combining the inequalities and we obtain _{(3) (4)}

Δ A

Δ A

≥ 1

2 ⟨[A

, A

]⟩

This inequality represents the general form of the Heisenberg uncertainty relation An important special case is

A

= x

A

= p

Δx

Δp

≥ ℏ

2 δ

24

Continuity equation

The interpretation of as probability density suggests that a probability current density should exist ^{ρ = | ψ (x, t) |}

2

We consider the case in which the Hamiltonian is

H = p

/2m + V(x)

and we compute the derivative of the probability density with respect to _t

∂ρ

∂t = ∂ψ*

∂t ψ + ψ * ∂ψ

∂t

Then we use the Schrödinger equation and its complex conjugate

iℏ(∂ψ / ∂t) = Hψ − iℏ(∂ψ */ ∂t) = (Hψ)*

to find _∂ρ

∂t = i

ℏ ( (Hψ)*ψ − ψ *Hψ )

The potential is real: the term in _V(x) cancels out and we obtain

25

Continuity equation

If we define _{j = −} ^iℏ

2m ( ψ * ∇ψ − ψ ∇ ψ * )

we find the continuity equation ^∂ρ

∂t + ∇ ⋅ j = 0

∂ρ

∂t = − iℏ

2m ( ( ∇

ψ*)ψ − ψ* ∇

ψ ) = iℏ

2m ∇ ⋅ ( ( ∇ ψ*)ψ − ψ * ∇ ψ )

in integral form it gives

d

dt ∫

d

x ρ(x, t) = − ∫

dS n ⋅ j(x, t)

The meaning of the continuity equation (as for the charge density in

electromagnetism) is that the dynamics cannot create or destroy particles

26

Stationary states and time evolution

We now go back to the Schrödinger equation

And we assume that the Hamiltonian is independent of time

iℏ ∂ ψ

∂t = Hψ

In this case a solution can be found by separation of variables _{ψ (x, t) = f (t)ψ (x)}

Plugging this ansatz in the Schrödinger equation and dividing by _{ψ (x, t)} we find

1

f (t) iℏ ∂ f (t)

∂t = 1

ψ (x) Hψ

Since the left-hand side depends only on and the right-hand side only on both sides must be equal to a constant that we call ^t _E ^x Thus for we obtain _{f (t) iℏ ∂} ^{f (t)}

∂t = E f (t) with solution f (t) = e ^−iEt/ℏ

Recall Lecture 2: this is exactly the same procedure used for the free particle

27

Stationary states and time evolution

Hψ (x) = E ψ (x)

The equation for _{ψ (x)} reads

Which is known as time-independent Schrödinger equation and it is an eigenvalue equation for the operator _H

The solutions _{ψ (x, t) = e}

_{ψ (x)} are called stationary states

Note that the associated probability density _{ρ(x, t) = | ψ (x) |}

is independent of time The solutions of Eq. provide a useful basis for the Hilbert space _{(5) {ϕ}

_}

(5)

Given an initial condition we can proceed as we did in Lecture 2 for the free particles: we expand it on the orthonormal basis ^{ψ (x, t = 0) = ψ}

^(x)

ψ

(x) = ∑

k

c

ϕ

(x) with _c

_{= ⟨ϕ}

_{| ψ}

_⟩ Then the solution of the

Schrödinger equation reads ^{ψ (x, t) = ∑}

^c

^e

−iE_kt/ℏ

ϕ

(x)

Hϕ _k = E _k ϕ _k

The Heisenberg picture

In classical mechanics there is only one way to describe the time evolution of a system through the time dependence of the observables

In quantum mechanics we can do it in (at least) two ways:

Suppose, as we did up to now, that the state depends on time, and thus it tells us how the probabilities of of observables measurements evolve

Assume that it is the operators associated to the observables to depend on time, and that the state remains fixed

The first possibility leads to the Schrödinger picture, while the second leads to the Heisenberg picture

What is physically observable are the matrix elements of operators

⟨A⟩

(t) = ⟨ψ

(t) | A

| ψ

(t)⟩

Where we have added to denote that we work in the Schrödinger picture _S

28

⟨A⟩

(t) = ⟨ψ

(t) | A

| ψ

(t)⟩ = ⟨ψ

(t

) | U

(t, t

)A

U(t, t

) | ψ

(t

)⟩

We can move to the Heisenberg picture by writing

29

Where the operator _{U(t, t}

₎ is the time evolution operator defined as

| ψ

(t)⟩ = U(t, t

) | ψ

(t

)⟩

And it obeys the Schrödinger equation _{iℏ ∂} ^U

∂t = H U

We can thus define the operator in the Heisenberg picture as _A

A

(t) ≡ U

(t, t

)A

U(t, t

)

The matrix element of the operator is clearly independent on which representation is used ^A

With _A

_(t

_{) = A}

and _{| ψ}

_{⟩ = | ψ}

_(t

_)⟩

30

The Ehrenfest Theorem

We have seen with the path integral approach how classical mechanics is obtained in the limit _{ℏ → 0}

Another perspective is offered by the Ehrenfest theorem

We start from the Schrödinger equation and its complex conjugate

iℏ(∂ψ / ∂t) = Hψ − iℏ(∂ψ */ ∂t) = (Hψ)*

We want to construct the dynamical equation for the expectation value of an operator _A

⟨A⟩ = ∫ d

x ψ *(x, t) A ψ (x, t)

d

dt ⟨A⟩ = ∫ d

x ( ∂ψ *

∂t A ψ + ψ* ∂A

∂t ψ + ψ * A ∂ψ

∂t )

Taking the derivative with respect to we obtain _t

31

dt d ⟨A⟩ = ∫ d

x ( i

ℏ (Hψ)* A ψ + ψ* ∂A

∂t ψ − i

ℏ ψ * AHψ )

By using the Schrödinger equation and its conjugate

we obtain ^∂

^{ψ = − i / ℏ(Hψ) ∂}

^{ψ* = i / ℏ(Hψ)*}

Since is Hermitian we have _H

d

dt ⟨A⟩ = i

ℏ ⟨[H, A]⟩ + ⟨ ∂ A

∂t ⟩

We can now compare with what we have in classical mechanics: for a system with conjugated variables and we have _{q p}

d

dt f (q, p, t) = ∂f

∂q q · + ∂f

∂p p · + ∂f

∂t = ∂f

∂q ∂H

∂p − ∂ f

∂p ∂H

∂q + ∂f

∂t = {H, f } + ∂f

∂t

where the Poisson brackets are defined as {g, f } = ∂g

∂p

∂f

∂q − ∂ f

∂p

∂g

∂q

We conclude that we

have the correspondence ^{{ f, g} ↔} _ℏ ^{i [F, G]}

Where the and are

observables and and are the corresponding operators

f g

F G

(6)

use Hamilton equations _q ^· _{= ∂H / ∂p} and _p ^· _{= − ∂H/ ∂q}

We now want to apply the theorem to obtain the time evolution of and _{⟨x⟩ ⟨p⟩}

We need

[H, x

] = ∑

j

p

2m , x

= 1

2m ∑

j

[ p

, x

] = − iℏ p

m

[H, p

] = iℏ ∂ V

∂x

d

dt ⟨p

⟩ = i

ℏ ⟨[H, p

]⟩ = −

⟨

∂V

∂x

⟩ d

dt ⟨x

⟩ = i

ℏ ⟨[H, x

]⟩ = ⟨p

⟩ m

These equations have the same form as their classical counterparts

use _[A

_{, B] = [A, B]A + A[A, B]}

The above equations (and more generally Eq. ) are known as the Ehrenfest theorem: “Classical equations hold for expectation values of quantum operators” ⁽⁶⁾

Thus we can say that classical mechanics emerges as the limit in which the uncertainty (that is the standard deviation in a given measurement) is small with respect to the typical value of the measured quantity

32

Wave packets

The momentum eigenfunctions

ϕ

(x) = (2π ℏ)

e

Are states of definite momentum, and thus, according to the uncertainty principle, in these states the position is completely undetermined

Indeed by squaring the above expression, we obtain a constant independent on _x In realistic situations the momentum is measured with a finite resolution, on a system which is localised over a finite region of space

A realistic free particle is thus in a state which is a

superposition of momentum eigenstates, that is a wave packet The most general wave packet can be constructed as

33

ψ (x, t) = 1 2π ∫ dkc(k)e

We now focus on a special case of wave packet, the one with minimal uncertainty We require that at _{t = 0} the product _ΔpΔx takes the minimum value

ΔpΔx = ℏ

To construct a wave packet with this property, we go back to the proof of the 2

uncertainty principle (slide 21-23)

| ⟨ψ | A ̂

A ̂

ψ⟩ |

= 1 4 | ⟨ψ | { A

̂ , A

̂ }ψ⟩ |

+ 1 4 | ⟨ψ | [ A

̂ , A

̂ ]ψ⟩ |

≥ 1

4 | ⟨ψ | [ A

̂ , A

̂ ]ψ⟩ |

Defining _A

^̂ _{= A}

_{− ⟨A}

_{⟩ = A}

_{− ⟨ψ | A}

_{| ψ⟩} we used the two inequalities

| ⟨ψ | A

̂ A

̂ ψ⟩ |

≤ ⟨ψ | A

̂ | ψ⟩⟨ψ | A ̂

| ψ⟩ (3)

and

(4)

We have to look for the situation in which the two inequalities become equalities

34

As for Eq. (4) it becomes an equality if

⟨ψ | { A

̂ , A

̂ } | ψ⟩ = 2Re⟨ψ | A ̂

A ̂

| ψ⟩ = 2Re⟨ ̂ A

ψ | A ̂

ψ⟩ = 0

| ⟨ ̂ A

ψ | A

̂ ψ⟩ |

≤ ⟨ ̂ A

ψ | A ̂

ψ⟩⟨ ̂ A

ψ | A ̂

ψ⟩

As for Eq.(3) we can rewrite it as (since the operators are Hermitian)

Which tells us that the equality holds if the vectors and are

proportional ^A

^{| ψ⟩ A}

^{| ψ⟩}

| A

̂ ψ⟩ = c | A ̂

ψ⟩

Combining the two conditions we conclude that the coefficient must be purely

imaginary ^c

c = iλ

c ∈ ℂ

λ ∈ ℝ

If we now set _A

_{= p} and _A

_{= x} we find

( p − ⟨ p⟩ ) | ψ⟩ = i λ ( x − ⟨x⟩ ) | ψ⟩

or −iℏ ∂ ψ (x)

∂x − p

ψ (x) = iλ(x − x

)ψ (x) _x

_{= ⟨x⟩, p}

_{= ⟨ p⟩}

35

∂ψ (x)

∂x = i

ℏ p

ψ (x) − λ

ℏ (x − x

)ψ (x)

This is a first order differential equation which can be rewritten as

whose solution is

ψ (x) = C exp { ip

x

ℏ − λ

2ℏ (x − x

)

}

We thus conclude that the state of minimum uncertainty is a Gaussian centred in and modulated by a plane wave which determines the expectation value of the

momentum

x

p

By normalising the wave function to unity we obtain _{C = (} ^λ

π ℏ )

1/4

36

We can now check that this is indeed a wave packet with minimum uncertainty Position uncertainty

We first check that _{⟨x⟩ = x}

We have

⟨x − x

⟩ = ∫

+∞

−∞

dx(x − x

) | C |

exp { − λ

ℏ (x − x

)

} = 0 ⟨x⟩ = x

because the function is odd

Δ

x = ⟨(x − x

)

⟩ = ∫

+∞

−∞

dx(x − x

)

| C |

exp { − λ

ℏ (x − x

)

}

The position uncertainty is

= ( λ ℏπ )

1/2

π

2 ( λ ℏ )

−3/2

Use _∫

−∞

y

dye

= π

2 a

= ℏ 2λ

Momentum uncertainty

We first check that _{⟨p⟩ = p}

⟨ p⟩ = ∫

+∞

−∞

dx ψ *(x)(− iℏ) ∂ψ (x)

∂x = ∫

+∞

−∞

dx ψ *(x)(− iℏ) ( ip

ℏ − λ

2ℏ 2(x − x

) ) ψ (x) = p

odd

The momentum uncertainty is

Δ

p = ⟨( p − p

)

⟩ = ∫

+∞

−∞

dx ψ*(x) ( − iℏ ∂

∂x − p

) ( − iℏ ∂

∂x − p

) ψ (x)

= ∫

+∞

−∞

dx ψ *(x) ( − iℏ ∂ ∂x − p

) (− iℏ) ( − λ

ℏ (x − x

) ) ψ (x)

37

Δ

p = ℏλ

2 Δ

x = ℏ

2λ so that indeed _{ΔpΔx =} ^ℏ

2

ψ (x) = C exp { ip

x

ℏ − (x − x

)

4Δ

x }

Expressing in terms of _{λ Δ}

_x we can write _{ψ (x)} as

and we see that really determines the width of the Gaussian ^Δx

= ∫

+∞

−∞

dx ψ *(x) ( − iℏ ∂ ∂x − p

) (− iℏ) ( − λ

ℏ (x − x

) ) ψ (x)

= ℏλ − λ

∫

+∞

−∞

dx | C |

(x − x

)

exp { − λ

ℏ (x − x

)

}

= ℏλ − λ

( λ ℏπ )

1/2

π

2 ( λ ℏ )

−3/2

= ℏλ

2 Using as before _∫

−∞

y

dye

= π

2 a

In summary we have

38

odd

Dropping as before an

odd term in _{(x − x}

₎

ψ (x, t) = 1 2π ∫ dkc(k)e

A general wave packet is

Each of the energy eigenstates moves with momentum and phase velocity given

by ^ℏk

v

= ℏk / m

The wave packet, however, can be considered as a single object, which moves with a single velocity, known as group velocity

v

= d

dt ⟨x⟩

The discussion of the Ehrenfest theorem shows that

v

= d

dt ⟨x⟩ = ⟨ p⟩

m

Such velocity can be interpreted as the velocity of the “centre” of the packet Phase and group velocity

39

Our gaussian wave packet moves

with group velocity _v

_{= p}

_{/ m}

Time evolution of a Gaussian packet

We have seen that the wave packet has minimum

uncertainty, where we have set the centre of the packet for simplicity and

ψ (x) = C exp { ip

x

ℏ − x

4Δ

x } x

= 0

We now want to study how the packet evolves with time

We can do it by using the free particle propagator (recall Lecture 1)

K(x, t, y,0) = m

2π iℏt exp { im(x − y)

2ℏt }

ψ (x, t) = ∫ dy K(x, t, y,0) ψ ( y,0)

= C ∫ dy m

2π iℏt exp { im(x − y)

2ℏt + ip

y

ℏ − y

4Δ

x } C = ( λ

πℏ )

1/4

= ( 2πΔ

x )

40

σ (t) = σ + iℏt 2m

The integral can be computed by using the standard trick of rewriting the exponent as

imx

2ℏt − ( 1

4Δ

x − im

2ℏt ) ( (y − y

)

− y

) y

= i 2ℏ

p

−

4Δ1²x

−

Gaussian integral

After some manipulations the result can be written in the following form

ψ (x, t) = ( σ

2πσ

(t) )

1/4

exp { i

ℏ ( p

x − p

2m t

) } exp { − (x − p

t / m)

4σ (t) }

where _{σ = Δ}

_x This is still a Gaussian packet but:

the momentum uncertainty remains constant: _Δ

_{p(t) = Δ}

_p(0) (this could also be expected from the Ehrenfest theorem) The center of the packet moves with velocity _p

_{/ m}

The position uncertainty increases with time _Δ

_{x(t) = Δ}

_x

( 1 + ℏ

t

4m

Δ

x )

41

Wrap up: the postulates of QM

1) The state of a system is represented by a vector (ket) _{| ψ⟩} in a Hilbert space _ℋ The specific Hilbert space depends on the system we consider (for example, for a one-dimensional particle in the interval the wave function

is a vector in the space of the square-integrable functions ^{[a, b]} )

ψ (x) ↔ | ψ⟩ L

[a, b]

The state vector _{| ψ⟩ ∈ ℋ} contains all the information of the system is the norm while is the scalar product in

∥ψ ∥ = ⟨ψ | ψ⟩ = 1 ⟨ ⋅ | ⋅ ⟩ ℋ

The phase of _{| ψ⟩} is not observable

42

We can now summarise what we have learnt and put it in the form of

postulates of the quantum theory

2) Every observable (position, momentum, angular momentum, energy…) corresponds to a self-adjoint (for us equivalent to hermitian) operator ^𝒜 _A

The adjoint operator is defined as _A

A self-adjoint operator is such that _{A = A}

⟨ψ | A χ⟩ = ⟨A

ψ | χ⟩ ≡ ⟨ψ | A | χ⟩

and _{⟨ψ | A | χ⟩* = ⟨ χ | A}

_{| ψ⟩}

For a self-adjoint operator we can write a spectral decomposition

A = ∑

k=1

a

| ϕ

⟩⟨ϕ

|

Where are the eigenvalues and the eigenvectors of , with and is the dimension of the Hilbert space (can be )

a

∈ ℝ | ϕ

⟩ A

A | ϕ

⟩ = a

| ϕ

⟩ N ℋ ∞

The eigenvectors _{| ϕ}

_⟩ form an orthonormal system for , i.e. _{ℋ ⟨ϕ}

_{| ϕ}

_{⟩ = δ}

43

3) The result of a measurement of an observable is an eigenvalue of the operator (more precisely a value in the spectrum of ) _A ^𝒜 _A

Note that (recall MMP1 !) every eigenvalue is in the spectrum, but not every element of the spectrum is an eigenvalue, because the spectrum can also be continuous

4) If the system is in the state , then the probability that a measurement of the observable gives the value is given by _𝒜 ^{| ψ⟩} _a

_P(a

_{) = | ⟨ϕ}

_{| ψ⟩ |}

If the eigenvalue is degenerate, we need to sum over all the corresponding _⟨ϕ

_| 5) Wave function collapse

When a system in the state undergoes a measurement of the observable which gives the result , the system will collapse in the state _a

^{| ψ⟩} _{| ϕ}

_⟩ ^𝒜 If the eigenvalue is degenerate, the system may collapse in any state in the eigenspace corresponding to _ℰ

_a

44

ℋ ∋ | ψ⟩ measurement of with result

| ϕ

⟩ ∈ ℰ

6) Time evolution

If at the time the system is in the state the state at time can be obtained through the evolution operator ^{t = t}

_U ^{| ψ (t}

^{)⟩ t}

The operator is unitary and fulfils the Schrödinger equation _U

| ψ (t)⟩ = U(t, t

) | ψ (t

)⟩

iℏ ∂ U

∂t = H U

The Hamilton operator is a self-adjoint operator corresponding to the energy. If is independent on time we have _H ^H

45

U(t, t

) = exp{− i

ℏ H(t − t

)}