Cossey L. Introduction to Mathematical Physics

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Heisenberg representation

We have seen that evolution operator provides state at time t as a function of state at time

φ(t) = Uφ(0)

Let us write φ

the state in Schr\"odinger representation and φ

the state in Heisenberg

representation. {Heisenberg representation} Heisenberg\footnote{Wener Heisenberg

received the Physics Nobel price for his work in quatum mechanics} representation is

defined from Schr\"odinger representation by the following unitary transformation:

φ = Vφ

H S

with

V = U

− 1

In other words, state in Heisenberg representation is characterized by a wave function

independent on t and equal to the corresponding state in Schr\"odinger representation for

t = 0 : φ = φ (0)

H S

. This allows us to adapt the postulate to Heisenberg representation:

Postulate: (Description of physical quantities) To each physical quantity and

corresponding state space can be associated a function with self

adjoint operators A (t)

in values.

Note that if A

is the operator associated to a physical quantity in Schr\"odinger

representation, then the relation between A

and A

is:

A (t) = U A U

Operator A

depends on time, even if A

does not.

Postulate: (Possible results) Value of a physical quantity at time t can only be one of the

points of the spectrum of the associated self adjoint operator A(t).

Spectral decomposition principle stays unchanged:

Postulate: (Spectral decomposition principle) When measuring some physical quantity

on a system in a normed state , the average value of measurements is < A >

< A > = < ψ | A ψ >

H H H

The relation with Schr\"odinger is described by the following equality:

< ψ | A ψ > = < ψ U | U A U | U ψ >

H H H S

As U is unitary:

< ψ | A ψ > = < ψ | A ψ >

H H H S S S

Postulate on the probability to obtain a value to measurement remains unchanged, except

that operator now depends on time, and vector doesn't.

Postulate: (Evolution) Evolution equation is (in the case of an isolated system):

This equation is called Heisenberg equation for the observable.

Remark: If system is conservative (H doesn't depend on time), then we have seen that

if we associate to a physical quantity at time t = 0 operator A(0) = A identical to operator

associated to this quantity in Schr\"odinger representation, operator A(t) is written:

Interaction representation

Assume that hamiltonian H can be shared into two parts H

and H

. In particle, H

is often

considered as a perturbation of H

and represents interaction between unperturbed states

(eigenvectors of H

). Let us note a state in Schr\"odinger representation and a

state in interaction representation.{interaction representation}

with

Postulate: (Description of physical quantities) To each physical quantity in a state space

is associated a function with self adjoint operators A (t)

in values.

If A

is the operator associated to a physical quantity in Schr\"odinger representation,

then relation between A

and A

is:

So, A

depends on time, even if A

does not. Possible results postulate remains

unchanged.

Postulate:

When measuring a physical quantity for a system in state ψ

, the average value of A

is:

< ψ | A ψ >

I I I

As done for Heisenberg representation, one can show that this result is equivalent to the

result obtained in the Schr\"odinger representation. From Schr\"odinger equation,

evolution equation for interaction representation can be obtained immediately:

Postulate: (Evolution) Evolution of a vector ψ

is given by:

with

Interaction representation makes easy perturbative calculations. It is used in quantum

electrodynamics . In the rest of this book, only Schr\"odinge representation will be used.

Some observables

Hamiltonian operators

Hamiltonian operator {hamiltonian operator} has been introduced as the infinitesimal

generator times of the evolution group. Experience, passage methods from classical

mechanics to quantum mechanics allow to give its expression for each considered

system. Schr\"odinger equation rotation invariance implies that the hamiltonian is a scalar

operator appendix chapgroupes).

Example:

Classical energy of a free particle is

Its quantum equivalent, the hamiltonian H is:

Remark: Passage relations Quantification rules

Position operator

Classical notion of position r of a particle leads to associate to a particle a set of three

operators (or observables) R ,R ,R

x y z

called position operators{position operator} and

defined by their action on a function φ of the orbital Hilbert space:

R φ(x,y,z) = xφ(x,y,z)

R φ(x,y,z) = yφ(x,y,z)

R φ(x,y,z) = zφ(x,y,z)

Momentum operator

In the same way, to "classical" momentum of a particle is associated a set of three

observables P = (P ,P ,P )

x y z

. Action of operator P

is defined by {momentum operator}:

Operators R and P verify commutation relations called canonical commutation relations

{commutation relations} :

[R,R ] = 0

i j

[P,P ] = 0

i j

where δ

is Kronecker symbol appendix secformultens) and where for any operator A

and B, [A,B] = AB − BA. Operator [A,B] is called the commutator of A and B.

Kinetic momentum operator

Definition:

A kinetic momentum {kinetic moment operator} J, is a set of three operators J ,J ,J

x y z

that

verify following commutation relations {commutation relations}:

that is:

where ε

ijk

is the permutation signature tensor appendix secformultens). Operator J is

called a vector operator appendix chapgroupes.

Example:

Orbital kinetic momentum

Theorem:

Operator defined by L = ε RP

i ijk j k

is a kinetic momentum. It is called orbital kinetic

momentum.

Proof:

Let us evaluate

Postulate:

To orbital kinetic momentum is associated a magnetic moment M:

Example:

Postulates for the electron. We have seen at section secespetat that state space for an

electron (a fermion of spin s = 1 / 2) is the tensorial product orbital state space and spin

state space. One defines an operator S called spin operator that acts inside spin state

space. It is postulated that this operator is a kinetic momentum and that it appears in the

hamiltonian {\it via} a magnetic momentum.

Postulate:

Operator S is a kinetic moment.

Postulate:

Electron is a particle of spin s = 1 / 2 and it has an intrinsic magnetic moment {magnetic

moment}:

Linear response in quantum mechanics

Let be the average of operator (observable) A. This average is accessible to the

experimentator ). The case where H(t) is proportional to sin(ωt) is treated in Case where

H(t) is proportional to δ(t) is treated here. Consider following problem:

Problem:

Find ψ such that:

with

and evaluate:

Remark: Linear response can be described in the classical frame where Schr\"odinger

equation is replaced by a classical mechanics evolution equation. Such models exist to

describe for instance electric or magnetic susceptibility.

Using the interaction representation\footnote{ This change of representation is equivalent

to a WKB method. Indeed, becomes a slowly varying function of t since temporal

dependence is absorbed by operator

}

and

Quantity

to be evaluated is:

At zeroth order:

Thus:

Now, has been prepared in the state ψ

, so:

At first order:

thus, using properties of δ Dirac distribution:

Let us now calculate the average: Up to first order,

Indeed,

is zero because Z is an odd operator.

where, closure relation has been used. Using perturbation results given by equation ---

pert1--- and equation ---pert2---:

We have thus:

Using Fourier transform\footnote{ Fourier transform of:

and Fourier transform of:

are different: Fourier transform of f(t) does not exist! ) }

Exercises

Exercice:

Study the binding between position operator R

and momentum operator P

with the

infinitesimal generator of the translation group

Exercice:

Study the connection between kinetic moment operator L and the infinitesimal generator

of the rotations group

Exercice:

Study the linear response of a classical oscillator whose equation is:

where f(t) is the excitation. Compare with the results obtained at section secreplinmq.

Introduction

In this chapter, N body problems that quantum mechanics can treat are presented. In

atoms, elements in interaction are the nucleus and the electrons. In molecules, several

nuclei and electrons are in interaction. Cristals are characterized by a periodical

arrangment of their atoms. In this chapter, only spectral properties of hamiltonians are

presented. Thermodynamical properties of collections of atoms and molecules are

presented at next chapter. From a mathematical point of view, this chapter is an

application of the spectral method presented at section chapmethspec to study linear

evolution problems.

Atoms

One nucleus, one electron

This case corresponds to the study of hydrogen atom.{atom} It is a particular case of

particle in a central potential problem, so that we apply methods presented at section ---

secpotcent--- to treat this problem. Potential is here:

It can be shown that eigenvalues of hamiltonian H with central potential depend in

general on two quantum numbers k and l, but that for particular potential given by

equation eqpotcenhy, eigenvalues depend only on sum n = k + l.

Rotation invariance

We treat in this section the particle in a central potential problem

. The spectral

problem to be solved is given by the following equation:

Laplacian operator can be expressed as a function of L

operator.

Theorem:

Laplacian operator ∆ can be written as:

Proof:

Here, tensorial notations are used (Einstein convention). By definition:

L = ε x p

i ijk j k

So:

The writing order of the operators is very important because operator do not commute.

They obey following commutation relations:

[x ,x ] = 0

j k

[p,p ] = 0

j k

From equation eqdefmomP, we have:

thus

Now,

Introducing operator: