Mathematical Formulation of Quantum Mechanics

Andreas WackerMathematical Physics

Lund University

Lund University / Science Faculty / Mathematical Physics / Andreas.Wacker@fysik.lu.se / 2015-01-21

Formulation of Classical Physics

Very successful in everyday lifeand forming our concept of nature

Objects, we observe, are described by points ri in the three-dimensional Euclidean space

Fully determined for given initial position and velocity

Time-evolution according to mi ri=Fi (Newton 1687)

Observation is not a matter of principle relevance

The classical concept fails

characteristic frequencies of light (Rydberg 1888 Lund)stability of atom (Rutherford 1911)Magic particle numbers and periodic table of elements

Black-body radiationSpecific heat of solidsPhysics of semiconductors (band gap!)Superconductivity

Features on atomic scale

Macroscopic effects

The Stern-Gerlach Experiment (1922)

Force ~ vertical component of the magnetic moment

Two discrete values μz=±μB observed

Drawn by Theresa Knott (wikipedia)

Formalism to describe quantum effects

Physical systems are described by elements of the ket space (generalizes wave functions)Measurement results become of stochastic natureMeasurement changes the physical system

Well defined formalism, which allows to do calculations and currently agrees with experiment

Interpretation is far from clear and continuously debated

Ket space as a vector space over the complex numbers

Commutativity of addition |ϕ1>+|ϕ2>=|ϕ2>+|ϕ1>

Associativity of addition (|ϕ1>+|ϕ2>)+|ϕ3>=|ϕ1>+(|ϕ2>+|ϕ3>)

(write +|ϕ> as -| >)ϕ1| >=| >ϕ ϕ(α+β)| >=α| >+β| > and α(|ϕ ϕ ϕ ϕ1>+|ϕ2>)=α|ϕ1>+α|ϕ2>

(αβ)| >=α(β| >)ϕ ϕ

kets | > and complex numbers α with two operationsϕAddition of two kets |ϕ1>+|ϕ2> is also a ket |ϕ3>

Multiplication with complex numbers α|ϕ1> is also a ket |ϕ2>

Satisfying:

The scalar (or inner) product and bra's

For |ϕ1> and |ϕ2> define inner product <ϕ1|ϕ2>=α with

< | > is real and positive, unless | >=|null>, where <null|null> =0ϕ ϕ ϕ<ϕ1|ϕ2>=<ϕ2|ϕ1>

For |ϕ3>=α|ϕ1>+β|ϕ2> we have

< |ϕ ϕ3>=α< |ϕ ϕ1>+β< |ϕ ϕ2>

<ϕ3| >=αϕ *<ϕ1| >+βϕ *<ϕ2| >ϕ

Treat < | as an object on its own, called ϕ bra.

The bras form their own vector space, the dual space

dual correspondence | > ϕ ⇆ < |ϕ

<ϕ1|ϕ2>bra(c)ket

Note |ϕ3>=α|ϕ1>+β|ϕ2> → <ϕ3|=α*<ϕ1|+β*<ϕ2|

A basis is a set {|ai>} which allows to construct any | > asϕ | >=ϕ ∑i ci|ai> with unique complex numbers ci

An orthonormal (ON) basis satisfies <ai|aj>=δij

For an ON basis we find

Expansion coefficients ci=<ai| >ϕ

Completeness relation ∑i |ai><ai|=1 ⚠< | >= ϕ ϕ ∑i |ci|

1. Any physical state is identified by a ket |ψ> with norm 1, i.e. <ψ|ψ>=1.

2. An ideal measurement provides real measurement values αn associated with physical states |an>, which form an ON basis of the ket space.

3. For an arbitrary state |ψ>, the probability to observe the result αn in a measurement is given by Pn= |<an|ψ>|2

Physical States and Measurements (Standard Copenhagen Interpretation)

After measuring the value αn the physical state collapses to the corresponding state |an>.

Fortuitousness is fundamental for quantum mechanics

Example: Stern-Gerlach Experiment

Force ~ vertical component of the magnetic moment

Two discrete values μz=±μB observed

Drawn by Theresa Knott (wikipedia)

Formulate on board

Linear Operators

Operator Â transforms Â|Φ>=|Φ'>

Linear operator: Â(α|Φ>+β|φ>)=αÂ|Φ>+βÂ|φ>

Product between operators Ĥ=ĈÂ the operator transforming Ĥ|Φ>=Ĉ(Â|Φ>)

Define commutator [Ĉ,Â]=ĈÂ-ÂĈ

Matrix representation

For ON basis {|ai>} evaluate

State | > ϕ → coefficients ci=<ai| >ϕOperator Â → coefficients Aij=<ai|Â|aj>

All calculations become matrix-operations

Note: depend on basis used, only for ON basis

∣φ ' ⟩= A∣φ ⟩⇔(c '1c '2⋮

)=(A11 A22 …

A21 A22 …

⋮ ⋮ ⋱)(

ϕ∣φ ⟩=(d1* d2

* …)(c1

The adjoint operator

Consider <φ|Â|Φ> for arbitrary |Φ>

<φ|Â| is a linear mapping |Φ>→complex numbers

Interpret <φ|Â|=<φ| as a new bra

How is the corresponding ket |φ> related to |φ>?

Relation linear⇒ There is an operator Ĉ with |φ>=Ĉ|φ>

As Ĉ depends on Â, write Ĉ=Â†, the adjoint operator of Â

Â†|φ> is dual correspondence of <φ|Â for all |φ>

Important relations for the adjoint operator

If Â†=Â: Â is called Hermitian (self-adjoint), Aij=Aji*

If Â†Â=1: Â is called unitary

φ2∣ A∣φ1 ⟩*= φ2∣φ1 ⟩*=φ1∣φ2 ⟩=φ1∣A†∣φ2 ⟩

In matrix representation ai∣A†∣a j ⟩=a j∣A∣ai ⟩

*=A ji*

Â†|φ> is dual correspondence of <φ|Â for all |φ>

(∣φ1 ⟩ φ2∣)†=∣φ2 ⟩ φ1∣

( A B)†=B† A† see exercise

( A†)†= A see exercise

(α A)†=α* A† see exercise

Hermitean Operators

Matrices with Aij=Aji* can be diagonalized

(A11 A12 …

A21 A22 …

⋮ ⋮ ⋱)(c1

⋮)=λn(

⋮) with real λn

Eigenstates ∣un ⟩=∑ic i

(n)∣ai ⟩ Α∣un ⟩=λn∣un ⟩with

form a new ON basis

Α Hermitean ⇔Α=∑n λn∣un ⟩ ⟨ un∣

If Â†=Â: Â is called Hermitean (self-adjoint), Aij=Aji*

Common eigenstates of two Hermitean operators Â and Ĉ

There exists an ON basis {|an>}, where all |an> are

simultaneously eigenstates of Â and Ĉ (possibly with different eigenvalues)

⇕[Â,Ĉ]=0

Note: [Â,Ĉ]=0 does not imply that each eigenstate of Â is also eigenstate of Ĉ

Operators representing observables

Measurement: Real measurement values αn associated with physical states |an> providing ON basis

Expectation value for state |Ψ>:

Physical observable ↔ Hermitean operator

Example: µz and µx of Stern Gerlach

⟨α ⟩=∑n

Pn⏟=∣⟨an∣Ψ ⟩∣2

αn=∑n

⟨ Ψ∣an ⟩αn ⟨an∣Ψ ⟩=⟨ Ψ∣Α∣Ψ ⟩

with Hermitean operator Α=∑n

∣an ⟩αn ⟨an∣

Phase of a quantum state

The states |Ψ> and |Ψ'>=eiα|Ψ> can not be distinguished by any measurement

The states |Ψ>=|Φ1>+|Φ2> and |Ψ'>=|Φ1>+eiα|Φ2> differ

Examples from Stern Gerlach

Continuous spectrum and spatial representation

For continuous measurement values, e.g. spatial position x

But continuum of states cannot be counted

Instead: ⟨ x∣x ' ⟩=δ(x−x ' ) and 1=∫dx∣x ⟩ ⟨ x∣

Eigenstates X∣x ⟩=x∣x ⟩ as before Α∣ai ⟩=αi∣ai ⟩

⟨a j∣ai ⟩=δ ji and 1=∑n

∣ai ⟩ ⟨ai∣ not possible

Any state can be written as ∣Ψ ⟩=∫dx∣x ⟩ ⟨ x∣Ψ ⟩

∣⟨ x∣Ψ ⟩∣2Δ x is probability to find position in interval (x , x+Δ x)

Common wave function Ψ(x)=⟨ x∣Ψ ⟩

Three spatial dimensions

Postulate [ X , Y ]=[ X , Z ]=[Y , Z ]=0

Wave function in 3 dimensions: Ψ(r)=⟨ x , y , z∣Ψ ⟩

⇒ There is a common set of eigenstates ∣x , y , z ⟩

for the operators X , Y , Z

Function of an operator ITaylor Series

Frequently we write f(Â) – what does it mean?

For f (x)=∑n=0∞

f nn!xn we define f (A)=∑n=0

∞f nn!

For f (x)=xn we define f (A)=An≡A A…⏟n times

Exercise: [B , An ]=n [B , A ] An−1 if [[B , A ] , A ]=0

Important implication:

[B , f (A )]=[B , A ] f ' ( A) if [[B , A ] , A ]=0

Function of an operator IIProjection on eigenstates

Hermitean operators have a complete set of eigenstates

A=∑n

αn |an><an |

f (A)=∑nf (αn)|an><an |Define

Example: ei A is unitary if A is Hermitean

Change of basis

Old basis {|an>}, new basis {|bn>}

Operator for basis change U=∑n

|bn><an | unitary!

| Ψ >=∑ncn |an>=∑

mc 'm |bm>

c 'm=⟨bm | Ψ⟩=∑n

⟨bm |an⟩⏟=⟨am | U† |an⟩

Matrix for basis change

Dynamical evolution of ket states:Hamiltonian

Simple systems with mechanical analog and potential V(x,y,z)

with the basic commutation relations

Common notation

Hamilton-operator Ĥ needs to be specified

iℏ ∂∂ t

∣Ψ(t)⟩=H∣Ψ(t) ⟩Define

2+ p y2 + pz

2m+V (r x , r y , rz)

[ p j , pk ]=0, [ r j , rk ]=0 , and [ p j , rk ]=ℏ

iδ jk

p= px ex+ p y e y+ pz e z and r=r x ex+ r y e y+ r z e z

Eigenstates of Hamiltonian

For time-independent Hamiltonian, search eigenstates

Arbitrary physical state:

provides time-dependence of system for given |Ψ(0)>

Stationary Schrödinger Eq. H∣φn ⟩=En∣φn ⟩

∣Ψ(t)⟩=∑ncn(t)∣φn ⟩ with cn(t)=cn(0)e

−iEn t / ℏ

Time-dependence of expectation values

ProofExample harmonic oscillator on board

Ψ∣Α∣Ψ ⟩=iℏ

Ψ∣[H , Α]∣Ψ ⟩+ Ψ∣∂ Α∂ t

∣Ψ ⟩

Spatial Representation

Consider the states |x> satisfying

In this basis we obtain the matrices

Need to show

Spatial representation of Schrödinger equation in 3 dimensions

For general operator

Solving the stationary Schrödinger equation by diagonalization

Search for eigen-energies En and states |φn>

Use matrix representation!

Stationary Schrödinger Eq. H∣φn ⟩=En∣φn ⟩

Summary

Any physical state is identified by a ket |ψ> with norm 1

i ℏ∂∣Ψ(t)⟩

∂ t=H ∣Ψ(t) ⟩Time-dependence by Hamilton operator

Observables are represented by Hermitian operators Â (Â†=Â)

The real eigenvalues αn are possible measurement values, which are found with probability Pn= |<an|ψ>|2 , for the corresponding eigenstates |an>

Unitary operators Û (Û†Û=1) describe the basis transformations between ON bases and keep the norm of kets

Average satisfies: ddt

Ψ∣Α∣Ψ ⟩=iℏ

Ψ∣[H , Α]∣Ψ ⟩+ Ψ∣∂ Α∂ t

∣Ψ ⟩

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