QM Review

Outline

• Postulates of QM• Expectation Values• Eigenfunctions & Eigenvalues• Where do we get wavefunctions from?

– Non-Relativistic– Relativistic

• Techniques for solving the Schro Eqn– Analytically– Numerically– Creation-Annihilation Ops

Postulates of Quantum Mechanics

• All information is contained in the wavefunction

• Probabilities are determined by the overlap of wavefunctions

• The time evolution of the wavefn given by

Hdt

di

2| ba

…plus a few more

Expectation Values

• Probability Density at r

• Prob of finding the system in a region d3r about r

• Prob of finding the system anywhere

)()( rr

rd 3

13 rdspaceall

• Average value of position r

• Average value of momentum p

• Expectation value of total energy

rdrspaceall

3

rdspaceall

3 p

rdspaceall

3 H

Eigenvalue Problems

Sometimes a function fn has a special property

fnthewrt

constsomefn

OpOp

eigenvalue eigenfn

Where do we get the wavefunctions from?

• Physics tools– Newton’s equation of motion– Conservation of Energy– Cons of Momentum & Ang Momentum

The most powerful and easy to use technique is Cons NRG.

Where do we get the wavefunctions from?

),(),(2

txtxV

ETot

P2

Non-relativistic: 1-D cartesian

KE + PE = Total E

)()(2

22

xExV totx

expexp

)(usually

decayingansionseriespower

or

edgessharpwithpotentialsforpieces

x

)()(2

xExV tot

2P

Where do we get the wavefunctions from?

)()(2

trtrV

ETot

P2

Non-relativistic: 3-D spherical

KE + PE = Total E

)()()(since trtr

)()(2

22

rErV tot

)()()(usually lmnl YrRr

)()(2

rErV tot

2P

02

)1(212

2

22

2

R

rVE

dr

dRr

dr

d

r tot

Non-relativistic: 3-D spherical

02

)1(22

2

22

2

u

rVE

dr

udtot

Most of the time set u(r) = R(r) / r

02

)1(212

2

22

2

R

rVE

dr

dRr

dr

d

r tot

02

)1(2"

2

2

2

u

rVEu tot

)()(

)( lmnl Yr

rurso But often

only one term!

Techniques for solving the Schro Eqn.

• Analytically– Solve the DiffyQ to obtain solns

• Numerically– Do the DiffyQ integrations with code

• Creation-Annihilation Operators– Pattern matching techniques derived from 1D SHO.

Analytic Techniques

• Simple Cases– Free particle (ER 6.2)– Infinite square well (ER 6.8)

• Continuous Potentials– 1-D Simple Harmonic Oscillator (ER 6.9, Table 6.1, and App I)– 3-D Attractive Coulomb (ER 7.2-6, Table 7.2)– 3-D Simple Harmonic Oscillator

• Discontinuous Potentials– Step Functions (ER 6.3-7)– Barriers (ER6.3-7)– Finite Square Well (ER App H)

Simple/Bare Coulomb

Eigenfns: Bare Coulomb - stationary statesnlm(r) or Rnl(r) Ylm()

-0.05

0

0.05

-0.05

0

0.05

-0.05

0

0.05

-0.05

0

0.05

-0.05

0

0.05

-0.05 0 0.05

-0.050

0.05

-0.2

0

0.2

-0.050

0.05

-0.1-0.05

00.05

0.1-0.1

-0.05

0

0.05

0.1

-0.04-0.02

00.020.04

-0.1-0.05

00.05

0.1

-0.1-0.0500.050.1

-0.1-0.05 00.050.1

-0.4

-0.2

0

0.2

0.4-0.1-0.05 00.050.1

-0.1-0.05 00.050.1

-0.1-0.050

0.050.1

-0.1-0.05

00.050.1

-0.1-0.05 00.050.1

-0.1-0.050

0.050.1

-0.1 00.1-0.100.1-0.04-0.0200.020.04

-0.1 00.1

http://asd-www.larc.nasa.gov/cgi-bin/SCOOL_Clouds/Cumulus/list.cgi

2,mY

Numerical Techniques

• Using expectations of what the wavefn should look like…– Numerical integration of 2nd order DiffyQ

– Relaxation methods

– ..

– ..

– Joe Blow’s idea

– Willy Don’s idea

– Cletus’ lame idea

– ..

– ..

ER 5.7, App G

SHO Creation-Annihilation Op Techniques

xmpim

a ˆˆ2

1ˆ

xmpi

ma ˆˆ

2

1ˆ

22

2

1

2

1

2

ˆ)( xk

m

paa H

Define:

ipx ˆ, 1ˆ,ˆ aa

If you know the gnd state wavefn o, then the nth excited state is:

ona ˆ

Inadequacy of Techniques

• Modern measurements require greater accuracy in model predictions.– Analytic– Numerical– Creation-Annihilation (SHO, Coul)

• More Refined Potential Energy Fn: V()– Time-Independent Perturbation Theory

• Changes in the System with Time– Time-Dependent Perturbation Theory