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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