schroedinger equation basic postulates of quantum...

Anna Lipniacka www.ift.uib.no/~lipniack/basic principles 20/01

Schroedinger equation

Basic postulates of quantum mechanics.

Operators: Hermitian operators, commutators

State function:eigenfunctions of hermitian operators-> normalization, orthogonality completeness

eigenvalues and expectation values of operators

Time independent Schroedinger equation and stationary states.

Probability current.


Schroedinger equation is a wave equation, which links time evolution of the wave function of the state to the Hamiltonian of the state. For most of systems Hamiltonian “represents” total energy of the system T+V= kinetic +potential.

Hamiltonian is defined also classically, and equations of motions forclassical systems can be written using derivatives of the Hamiltonian.

Classically there is no need for a concept of the wave function of the state, as any state can be totally specified by giving momentum and positions of allparticles.

Schroedinger equation.


- First I remind you about a flat wave . This is the wave function describinga free particle.- I will show that the flat wave is a solution of the free Schroedinger equation.

- It useful to “test” operators and properties of wave functions on a flat wave to understand what they really mean.


H=[p2

2 mV r ]

A non-relativistic particle has the following Hamiltonian-> energy:

where kinetic energy is E k=mv2

2=p2

2 m

r , t =2h−3

2 ∫pei

hpr−Et

d 3 p

we know that a free particle ( propagating in a place without potential) canbe described by a flat wave or a combination of flat waves- wave packet

m r , t =Ae

i

hpr−Et

where h=E ,=h / p ,ℏ=h

2

we note that:

∂r , t ∂ t

=2h−3

2 −i

h∫pE e

i

hpr−Et

d 3 p


Thus

∂r , t ∂ t

=2h−3

2 −i

h∫p p2

2 me

i

hpr−Et

d 3 p

Now remind ourselves Laplace operator

∇ 2= ∂2

∂ x2 ∂2

∂ y2 ∂2

∂ z2 and note that for example :

∂2r , t

∂ x2=−2ℏ

−3

2 1

ℏ2∫p px

2 ei

ℏpr−Et

d 3 p

∇ 2r , t =−2ℏ−3

2 1

ℏ2∫p p2 e

i

ℏpr−Et

d 3 p

i h∂∂ t=[−

h2

2 m∇ 2 ]

thus

This is Schroedingerequation for free particle


i h∂∂ t= H=[

p2

2 mV r ]

i h∂∂ t=[−

h2

2 m∇ 2V r ]

We can also note that differentiating over x (for example):

h

i

∂r , t ∂ x

=−2h−3

2 ∫p px e

i

hpr−Et

d 3 p

and define momentum operators px≝

h

i

∂∂ x

, py≝h

i

∂∂ y

, pz≝h

i

∂∂ z

i h∂∂ t=[−

h2

2 m∇ 2 ] i h

∂∂ t= H=[

p2

2 m]

Natural extension to the situation with potential ( non-free particle)


A ) For every “classical” observable a linear operator. It can depend on momentum and positionoperators. Momentum operator (x) is proportional to derivative over x

B) A system is fully described by a wave-function which fulfills wave equation, Schroedingerfor non-relativistic Hamiltonian

i h∂∂ t= H=[

p2

2 mV r ]

C) expectation value of an operator F

corresponds to the expectation ( mean value) of the measurement result of the variable Ftaken over a big number of independent measurements (this is more difficultthen it sounds)D) the only possible results of single measurements of the variable F are the eigenvaluesof the operator F F= f (example of spin ½ , z projections )

Postulates of Quantum Mechanics

F q1, q2, .. qn , p1,p2, .. pn: qn=qn , pn=

ℏi∂∂q

n

f = constant

< F >=∫* F d

(after a measurement the system “collapses” to the state with well defined variable f)


What's new here ?

You have heard this before on “kvantefysikk og statistik mekanikk”

The difference is that now we are trying to write our wave function ina more general way. It can be a function of space variables, or momentumor perhaps even more general variables ( and time).In practice we will start with space representation, then we will discussmomentum representation, and then other- general Dirac representationof the sate and vector representation ( matrix mechanics)


Ad C.

< F >=∫* F d d =dq1 .... d qn

In general we integrate over the set variables the wave function of the system isdefined on, and normalized on. In “space representation” of quantum mechanicswe discuss right now this is space coordinates. For example an expectationvalue of a position of a particle will be:

<r t >=∫∫∫*r , t rr , t d 3 r

<r t >=∫∫∫r ∣r , t ∣2 d 3 rwhat about momentum ?

1=∫∫∫∣r , t ∣2 d 3 r“proper normalization”

This is probability density to find the state in location r


Definition of hermitian operator: The operator F is hermitian if :

∫1* F2 d =∫2 F1

* d If the operator is hermitian its expectation value is REAL- that is a requirement foroperators associated with observables

∫1* F 2 d =∫2 F1

* d

Definition of adjoined operator to F

Hermitian operators are self-adjoined meaning : F = F

Hermitian operators.

The expectation values of an operator representing any real variable must be real

< F >=∫* F d =∫ F* d =< F >*

example, check p_xoperator.


Ad hermitian, prove of hermeticity condition

< F >=∫* F d =∫ F* d

∫1eia2* F 1eia2d =∫1eia2 F1eia F2

* d

∫ 1* F1 d eia∫1

* F2e−ia∫2* F1∫ 2

* F2 d =

=∫1 F1* d e−ia∫1 F2

*eia∫2 F1*∫2

F2* d

(alpha, any constant number- take real functions PSI1, PSI2

∫1* F2 d =∫2 F1

* d

∫2* F1 d =∫1 F2

* d


Commutator of operators

Commutators: are important! for example variables who's operators commutecan be measured simultaneously for the system.

[ A , B ]= A B− B ALets check [ x , p

x]= x px

− px x

x px− px x=xℏi

∂∂ x−ℏi

∂ x∂ x

= xℏi

∂∂ x−ℏi

x∂∂ x−ℏi=ℏ i

[ x , px ]=i ℏx position variable, and p_x donot commute.What does that mean forsubsequent measurements of px and x ?


Eigenfunctions and eigenvalues:

The result of an operator working on a function is usually a different function . If there exista set of functions that : Fn= f nn

we call them eigenfunctions of the operator F, and fn are eigenvalues (constants)

The set of eigenvalues ( or spectrum of the operator) can be discrete , continuos or mixed.Example of continuos- energy operator (Hamiltonian) for a free particle, discrete= hamiltonianfor harmonic oscillator, mixed- hamiltonian for hydrogen atom.

px=ℏi

∂∂ x

ℏi

∂∂ x f= f f f=const∗exp

ifx

ℏ

That's the form of eigenfunctions of momentum operator. They are not quadraticalyintegrable and have to be normalized in a different way.


Expectation value vs. eigenvalue of an operator.

The expectation ( mean value) of the measurement of the variable Ftaken over a big number of independent measurements ( in practice over large numberof identically prepared states- an ensamble ).

But the only possible results of measurements of the variable F are the eigenvaluesof the operator F

F= f (example of spin ½ , z projections )

< F >=∫* F d

If the system is in the state described by the eigenfunction of the operator ,the expectationvalue of the measurement is equal to the eigenvalue. As the eigenvalueis the only possible measurement results, that is what we will always get !

< F >=∫* F d =∫n

* Fnd = f

n∫n

*n= f

n

This proof is straightforward for normalized quadratic-integrable eigenfunctions,can be also proved for different type of normalization

for the eigenstate number “n”


Orthogonality of eigenfunctionsFor a hermetic operator, eigenfunctions corresponding to different eigenvaluesare orthogonal∫n

* Fm d =∫m Fn* d

f m∫n

*m d = f n

*∫mn

* d

fm− f

n∫n

*m

d =0

if f m≠ f n then ∫n

*m d =0

Scalar productDirac notation:

∫n

*m d =⟨n∣m⟩ ∫n

* Fm d =⟨n∣F m⟩

⟨n∣F m⟩=⟨ F n∣m⟩ Hermitian operator in Dirac notation


We try to normalize our set of eigenfunctions of an operator – typicallychoosing a constant to multiply the functions with.

∫n

*m d =nm0 for n = m , orthogonality1 for n=m , normalization

Normalization of eigenfunctions with continuous spectrum of eigenvalues has to be a bit different- functions are not “quadraticallyintegrable”

∫ f

* 'fd = f '− f =< f' | f>

Dirac delta f x =c∗exp ifx

ℏ

Example momentum eigenfunctions, here just 1-dim

∫ f

* ' x f x d x=1

2ℏ∫−∞∞

eix f− f ' ℏ dx= f '− f

C must be c=1

2ℏto get the normalization right


Dirac delta function :∫−∞

∞

f x x−x ' dx= f x ' ∫−∞

∞

f x x dx= f 0

properties : ∫−∞

∞

1∗x dx=1

possible representation:

x =lim 0x =lim

1

2∫−∞

∞

exp ixy−∣y∣dy=lim 0

x22

x =1

2∫−∞

∞

exp ixy dy

More about Dirac delta function:


x =

x22

x

=0.01

=0.1

=0.5


For eigenfunctions with continuous spectrum of eigenvalues we can represent a function g in the following way : (eigenfunctions have to form a complete basis set)g=∫ c f f df | g >=∫ df c f | f >=∫ df <f| g> | f >

∫ f

* 'fd = f '− f =< f' | f> then we have :

c f =∫ f

* g d =< f | g>

we must also have : ∫ f

* r ' f r d f =r '−r

Completeness We assume that every “reasonable” , quadratically integrable function canbe expressed as a linear combination of eigenfunctions of a hermitian operatorF g=∑

i

c ii | g >=∑i

c i | i >=∑i

<i | g> | i >

for orthonormal set of eigenfunctions the coefficients are a scalar productof the function in question and appropriate eigenfunction

∫n

* g d =∑i

c i∫n

*i d =cn cn=∫n

* g d =<n | g>


Completeness, continuation: Lets now consider that our functionsare normalized in the normal “space”, so

∫ f

* 'fd =∫ f

* ' r fr d 3 r= f '

x− f

x f '

y− f

y f '

z− f

z

etc..

g r =∑i

c iir cn=∫n

* r ' g r ' d 3 r '

g r =∫ g r ' ∑i

i

*r ' ir d3 r '

∑i

i

*r ' ir =r−r '

we must in analogyhave for continuousspectrum:

∫ f

* r ' f r d f =r '−r

3-dim Diracdelta


What is the interpretation of expansion coefficients cn ? We see that modulussquared of an expansion coefficient will correspond to a probabilityto measure certain eigenvalue: Lets take spectrum of eigenfunctionsof given operator F and expand a function of state (g) into it :

g=∑i

c ii and check what is the expectation value of the operatorF for the state described by g

< F >=∫ g* F g d =∫ ∑n

cn

*n

* F ∑i

c ii

< F >=∫ ∑n

cn

*n

*∑i

f i c ii=∑n

∑i

cn * c i f i∫n

*i d

< F >=∑n

∑i

cn* c

if

ii,n=∑

i∣c i∣

2f

i

but we know that from interpretation of expectation value we must have

< F >=∑i

P i f iwhere P_i is the probability that the value f_i will be measured.


Thus we have.

The probability that measuring observable associated with F on a statedescribed by a wave function “g” will give a result f_i is the following:

P i=∣c i∣2=∣∫i

* g d ∣2 whereF

i= f

i

i

For continuos spectrum of F we can prove in analogy that:

⟨F ⟩=∫ f ∣c f ∣2 df

The probability that measuring observable associated with F on a statedescribed by a wave function “g” will give a result beetween f and f+df is

P f df =∣c f ∣2 df =∣∫ f

* g d ∣2 df where F f= f f


Stationary states:If the Hamiltonian does not contain time explicitly we can try to separate the solutionsin to time dependent part and coordinates dependent part. We obtain that there isnew constant involved proportional to the time derivative of the time dependentfunction divided by the function itself. We call it ENERGY. We obtain time-independentShroedinger equation for the part which does not depend on time.

t ,r ≝r T t

i ℏdT t /dt

T t =const≝E

Hr =Er T=C e−i Et /ℏ

What “stationary” means in practice ? Expectation values of operatorsdo not depend on time. ( for normal operators which do not containtime derivatives ) Prove it !

i h∂T t ∂ t

r =[−h2

2 m∇ 2r V r r ]T t


∂r , t ∂ t

=∂*∂ t

=* ∂∂ t

∂*

∂ t=i

ℏ2 m

*∇ 2−∇ 2*=−iℏ

2 m∇∇*−*∇

change of probability density with time (at a given place)is related to the out-flow of the current.

∂∂ t=−i

ℏ2 m

∇∇*−*∇=−∇j

j=iℏ

2 m∇*−*∇=ℜ* i

ℏm∇

Probability interpretation for the particle wave function: modulus square of it is a probabilityto find a particle in a given place (probability density). THUS: Integral over space has to give 1. However locally spacial probability density can changewith time. We can define the probability current, useful when discussing movement of particles

Probability current :

i h∂∂ t=[−

h2

2 m∇ 2V r ] i h

∂*

∂ t=[−

h2

2 m∇ 2V r ]*

schroedinger equation basic postulates of quantum...

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