Transcript ppt

QM Reminder
C Nave @ gsu.edu
http://hyperphysics.phy-astr.gsu.edu/hbase/quacon.html#quacon
Outline
• Postulates of QM
• Picking Information Out of Wavefunctions
– Expectation Values
– Eigenfunctions & Eigenvalues
• Where do we get wavefunctions from?
– Non-Relativistic
– Relativistic
• What good-looking Ys look like
• Techniques for solving the Schro Eqn
– Analytically
– Numerically
– Creation-Annihilation Ops
Postulates of Quantum Mechanics
• The state of a physical system is completely described by a
wavefunction Y.
• All information is contained in the wavefunction Y
• Probabilities are determined by the overlap of
wavefunctions
 Ya | Yb 
2
Postulates of QM
• Every measurable physical quantity has a corresponding operator.
• The results of any individ measurement yields one of the
eigenvalues ln
of the corresponding operator.
• Given a Hermetian Op with eigenvalues ln and eigenvectors Fn ,
the probability of measuring the eigenvalue ln is
F
*
n
Y d r
3
2
or
Fn Y
2
Postulates of QM
• If measurement of an observable gives a result ln , then
immediately afterward the system is in state fn .
• The time evolution of a system is given by
• .
d
i Y  H Y
dt
corresponds to
classical Hamiltonian
Picking Information out of
Wavefunctions
Expectation Values
Eigenvalue Problems
Common Operators
• Position
r = ( x, y, z )
- Cartesian repn
• Momentum
p   i    i (  x ,  y ,  z )
• Total Energy
E
 i  t
• Angular Momentum
op
tot
L=rxp
- work it out
Using Operators: A
• Usual situation: Expectation Values
A


Y * A Y d 3r
all space
• Special situations: Eigenvalue Problems
AY  l Y
the original wavefn
a constant
(as far as A is concerned)
Expectation Values
• Probability Density at r
Y  (rf ) Y(rf )
• Prob of finding the system in a region d3r about r
Y
Y d 3r
• Prob of finding the system anywhere

Y
all space
Y d 3r
 1
• Average value of position r

Y


r Y d 3r
all space
• Average value of momentum p

Y


p Y d 3r
all space
• Expectation value of total energy

Y
all space
H Y d 3r
Eigenvalue Problems
Sometimes a function fn has a special property
Op
 some const 

fn  
 wrt the Op 
eigenvalue
fn
eigenfn
Since this is simpler than doing integrals, we usually label QM systems
by their list of eigenvalues (aka quantum numbers).
Eigenfns: 1-D Plane Wave moving in +x direction
Y(x,t) = A sin(kx-wt) or A cos(kx-wt) or A ei(kx-wt)
• Y is an eigenfunction of Px
Px Y   i x ei(kxw t ) 
k ei ( kx w t )
 k Y
• Y is an eigenfunction of Tot E
Tot E
Y  i t ei ( kx w t )
 w ei ( kx w t )
• Y is not an eigenfunction of position X
XY
 x ei ( kx w t )
 xY
 w Y
Eigenfns: Hydrogenic atom
•
Y is an eigenfunction of Tot E
Tot E
•
 P2


Ynlm (rf )  H Ynlm (rf )  
 V  Ynlm (rf )
 2m

mZ 2 e 4
1
Z2
 
Ynlm (rf )   2 13.6 Ynlm (rf )
2
2
2
(4o ) 2 n
n
Y is an eigenfunction of
L2 Ynlm (rf )
L z Ynlm (rf )
•
Ynlm(r,,f)
L2
and Lz
 (  1) 2 Ynlm (rf )
 m 2 Ynlm (rf )
Y is an eigenfunction of parity
Parity Ynlm (rf )  () Ynlm (rf )
units eV
Eigenfns: Hydrogenic atom
Ynlm(r,,f)
• Y is not an eigenfn of position X, Y, Z
• Y is not an eigenfn of the momentum vector Px , Py , Pz
• Y is not an eigenfn of
Lx and Ly
Where Wavefunctions come from
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.
Schrödinger Wave Equation
Use non-relativistic formula for Total Energy Ops
H
 KE  V

p2
 V
2m
and
op
Etot
Y(r, t )  i  t
Y(r, t )
 p2

 V  Y(r, t )  i  t

 2m

Y(r, t )
 2 2

  V  Y(r, t )  i  t

 2m

Y(r, t )
H
 i  t
http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians
Klein-Gordon Wave Equation
Start with the relativistic constraint for free particle:
Etot2 – p2c2 = m2c4 .
p2 = px2 + py2 + pz2
[ Etot2 – p2c2 ] Y(r,t) = m2c4 Y(r,t).
 (i  )
2
t
 ( i ) c 2
2

Y(r, t )  m2c 4
 a Monster to solve
Y(r, t )
Dirac Wave Equation
Wanted a linear relativistic equation
Etot2 – p2c2 = m2c4
p = ( px , py , pz )
[ Etot2 – p2c2  m2c4 ] Y(r,t) = 0
Change notation slightly

op
p0  Etot
/ c  i t
c
P4 = ( po , ipx , ipy , ipz )
~
[P42c2  m2c4 ] Y(r,t) = 0
difference of squares can be factored ~ ( P4c + mc2) (P4c-mc2)
and there are two options for how to do overall +/- signs
 4 coupled equations to solve.
Time Dependent Schro Eqn
d
i Y  H Y
dt
Where H = KE + Potl E
Y(x, t )
ER 5-5
Time Dependent Schro Eqn
d
i Y  H Y
dt
Where H = KE + Potl E
 p2

 V  Y(r, t )  i  t

 2m

Y(r, t )
 2 2

  V  Y(r, t )  i  t

 2m

Y(r, t )
Y(x, t )
Time Independent Schro Eqn
KE involves spatial derivatives only
If Pot’l E not time dependent, then Schro Eqn separable
Y(x, t )   (x)
f (t )
ref: Griffiths 2.1
Y(x, t )   (x ) e
 iEt / 
 2 2

  V  Y(r, t )  i  t

 2m

 2 2

  V  Y(r )  Etot

 2m

Y(r, t )
Y(r )
Drop to 1-D for ease
 2 2

 x  V (x ) Y(x )  Etot

 2m

Y(x )
ER 5-6
What Good Wavefunctions Look
Like
Sketching Pictures of Wavefunctions
Prob ~ Y* Y
KE
KE
+ V
=
Etot
 V (x ) Y(x )  Etot
Y(x )
 p2

 V (x ) Y(x )  Etot

 2m

Y(x )
 2 2

 x  V (x ) Y(x )  Etot

 2m

Y(x )
Bad Wavefunctions
Sketching Pictures of Wavefunctions
 2 2

 x  V (x ) Y(x )  Etot

 2m

Y(x )
To examine general behavior of wave fns, look for soln of the form
Y 
Ae
ik x
where k is not necessarily a constant
(but let’s pretend it is for a sec)
 2k 2
2m
k

 V
 Etot
2m
(Etot  V )
2

KE
Y 
k

Ae
ik x
2m
(Etot  V )
2

KE 
KE +
If Etot > V, then k Re
If Etot < V, then k Im
Y ~ kinda free particle
Y ~ decaying exponential
2/k ~ l ~ wavelength
1/k ~ 1/e distance
Sample Y(x) Sketches
•
•
•
•
Free Particles
Step Potentials
Barriers
Wells
Free Particle
Energy axis
V(x)=0 everywhere
1-D Step Potential
1-D Finite Square Well
1-D Harmonic Oscillator
1-D Infinite Square Well
1-D Barrier
NH3 Molecule
E&R Ch 5 Prob 23
Discrete or Continuous Excitation Spectrum ?
E&R Ch 5,
Prob 30
Which well goes with wfn ?
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)
Eigenfns: Bare Coulomb - stationary states
Ynlm(r,,f) or Rnl(r) Ylm(,f)
Simple/Bare
Coulomb
Numerical Techniques
ER 5.7, App G
• 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
..
..
SHO Creation-Annihilation Op
Techniques
Define:
aˆ 
1
2mw
aˆ 
( ipˆ  mw xˆ )

1
( ipˆ  mw xˆ )
2mw
 aˆ, aˆ   1
 x, pˆ   i

H  w ( a  a  12 ) 
pˆ 2 1 2
 kx
2m 2
If you know the gnd state wavefn Yo, then the nth excited state is:
(aˆ )

n
Yo
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