Schrodinger equation in QM Reminders.ppt

1. QM Reminder

2. C Nave @ gsu.edu
http://hyperphysics.phy-astr.gsu.edu/hbase/quacon.html#quacon

3. 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

4. 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
2
| 
Y
Y
 b
a

5. 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
2
2
3
*
Y
F
Y
F
 n
n or
r
d

6. 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
• . Y

Y H
dt
d
i
corresponds to
classical Hamiltonian

8. Picking Information out of
Wavefunctions
Expectation Values
Eigenvalue Problems

9. Common Operators
• Position
• Momentum
• Total Energy
• Angular Momentum
r = ( x, y, z ) - Cartesian repn
)
,
,
( z
y
x
i
i 






 

p
t
op
tot i
E 
 
L = r x p - work it out

10. Using Operators: A
• Usual situation: Expectation Values
• Special situations: Eigenvalue Problems
r
d
A
A
space
all
3
*
Y
Y
 
Y

Y l
A
the original wavefn
a constant
(as far as A is concerned)

11. Expectation Values
• Probability Density at r
• Prob of finding the system in a region d3r about r
• Prob of finding the system anywhere
)
(
)
( f
f r
r Y
Y
r
d3
Y
Y
1
3

Y
Y
 r
d
space
all

12. • Average value of position r
• Average value of momentum p
• Expectation value of total energy
r
d
r
space
all
3
Y
Y


r
d
space
all
3
Y
Y
 p

r
d
space
all
3
Y
Y
 H

13. Eigenvalue Problems
Sometimes a function fn has a special property
fn
the
wrt
const
some
fn 








Op
Op
eigenvalue
eigenfn
Since this is simpler than doing integrals, we usually label QM systems
by their list of eigenvalues (aka quantum numbers).

14. 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
• Y is an eigenfunction of Tot E
• Y is not an eigenfunction of position X
Y





Y 

k
e
k
e
i
x
t
kx
i
t
kx
i
x 

 )
(
)
( w
w
P
Y




Y 

w
w w
w


 )
(
)
( t
kx
i
t
kx
i
t e
e
i
E
Tot
Y


Y 
x
e
x t
kx
i )
( w
X

15. Eigenfns: Hydrogenic atom Ynlm(r,,f)
• Y is an eigenfunction of Tot E
• Y is an eigenfunction of L2 and Lz
• Y is an eigenfunction of parity
)
(
6
.
13
)
(
1
2
)
4
(
)
(
2
)
(
)
(
2
2
2
2
2
4
2
f
f

f
f
f
r
n
Z
r
n
e
mZ
r
V
m
r
r
nlm
nlm
o
nlm
nlm
nlm
Y


Y


Y










Y

Y

2
P
H
E
Tot
)
(
)
1
(
)
( 2
f
f r
r nlm
nlm Y


Y 


2
L
units eV
)
(
)
( 2
f
f r
m
r nlm
nlm
z Y

Y 
L
)
(
)
(
)
( f
f r
r nlm
nlm Y


Y 
Parity

16. 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

18. Where Wavefunctions come from

19. 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.

20. Schrödinger Wave Equation
V
m
V
KE
H 



2
2
p
Use non-relativistic formula for Total Energy Ops
t
op
tot i
E 
 
and
( ) ( )
t
i
t
H t ,
, r
r Y


Y 
( ) ( )
t
i
t
V
m
t ,
,
2
2
r
r
p
Y


Y






 
( ) ( )
t
i
t
V
m
t ,
,
2
2
2
r
r Y


Y








 

http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians

21. Klein-Gordon Wave Equation
Start with the relativistic constraint for free particle:
Etot
2 – p2c2 = m2c4 .
[ Etot
2 – p2c2 ] Y(r,t) = m2c4 Y(r,t).
( ) ( )
  ( ) ( )
t
r
c
m
t
r
c
i
i t ,
, 4
2
2
2
2
Y

Y



 

p2 = px
2 + py
2 + pz
2
 a Monster to solve

22. Dirac Wave Equation
Wanted a linear relativistic equation
[ Etot
2 – p2c2  m2c4 ] Y(r,t) = 0
Etot
2 – p2c2 = m2c4
Change notation slightly
t
op
tot
c
i
c
E
p 



/
0
p = ( px , py , pz )
~ [P4
2c2  m2c4 ] Y(r,t) = 0
P4 = ( po , ipx , ipy , ipz )
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.

23. Time Dependent Schro Eqn
Y

Y H
dt
d
i
Where H = KE + Potl E
( )
t
x,
Y

24. Time Dependent Schro Eqn
Y

Y H
dt
d
i
Where H = KE + Potl E
( )
t
x,
Y
( ) ( )
t
i
t
V
m
t ,
,
2
2
r
r
p
Y


Y






 
( ) ( )
t
i
t
V
m
t ,
,
2
2
2
r
r Y


Y








 

ER 5-5

25. Time Independent Schro Eqn
KE involves spatial derivatives only
If Pot’l E not time dependent, then Schro Eqn separable
( ) ( ) ( )
t
f
x
t
x 

Y ,
( ) ( ) 
/
, iEt
e
x
t
x 

Y 
ref: Griffiths 2.1

26. ( ) ( )
t
i
t
V
m
t ,
,
2
2
2
r
r Y


Y








 

( ) ( )
r
r Y

Y








 tot
E
V
m
2
2
2

( ) ( ) ( )
x
E
x
x
V
m
tot
x Y

Y









2
2
2

Drop to 1-D for ease

28. What Good Wavefunctions Look
Like
ER 5-6

29. Sketching Pictures of Wavefunctions
( ) ( ) ( )
x
E
x
x
V
m
tot
x Y

Y









2
2
2

( ) ( ) ( )
x
E
x
x
V
m
p
tot Y

Y







2
2
( )
  ( ) ( )
x
E
x
x
V
KE tot Y

Y

KE + V = Etot
Prob ~ Y* Y

30. Bad Wavefunctions

31. ( ) ( ) ( )
x
E
x
x
V
m
tot
x Y

Y









2
2
2

To examine general behavior of wave fns, look for soln of the form
x
ik
e
A

Y
where k is not necessarily a constant
(but let’s pretend it is for a sec)
tot
E
V
m
k


2
2
2

( )
V
E
m
k tot 
 2
2

Sketching Pictures of Wavefunctions
KE

32. ( )
V
E
m
k tot 
 2
2

x
ik
e
A

Y
If Etot > V, then k Re
Y ~ kinda free particle
If Etot < V, then k Im
Y ~ decaying exponential
2/k ~ l ~ wavelength 1/k ~ 1/e distance
KE + KE 

33. Sample Y(x) Sketches
• Free Particles
• Step Potentials
• Barriers
• Wells

34. Free Particle
Energy axis
V(x)=0 everywhere

35. 1-D Step Potential

37. 1-D Finite Square Well

38. 1-D Harmonic Oscillator

39. 1-D Infinite Square Well

40. 1-D Barrier

41. NH3 Molecule

42. E&R Ch 5 Prob 23
Discrete or Continuous Excitation Spectrum ?

43. E&R Ch 5,
Prob 30
Which well goes with wfn ?

46. 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.

47. 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)

50. Simple/Bare
Coulomb
Eigenfns: Bare Coulomb - stationary states
Ynlm(r,,f) or Rnl(r) Ylm(,f)

51. 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

52. SHO Creation-Annihilation Op
Techniques
( )
x
m
p
i
m
a ˆ
ˆ
2
1
ˆ w
w



( )
x
m
p
i
m
a ˆ
ˆ
2
1
ˆ w
w





2
2
2
1
2
1
2
ˆ
)
( x
k
m
p
a
a 


 
w

H
Define:
  
i
p
x 
ˆ
,   1
ˆ
,
ˆ 

a
a
If you know the gnd state wavefn Yo, then the nth excited state is:
( ) o
n
a Y

ˆ

53. 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