Download Physics - Mike Peel

Equations – Atomic Physics
Energies
E  hf  
E 2  p2 c2  m2 c 4
Potential energy between two charges:
qq
V r    1 2
4 o r
Potential due to Angular Momentum:
  1 2
V r  
2mr r 2
Rydberg Energy:
2
 e  mr

8 o ao '  4 o  2 2
Photoelectric effect ( W  Work function):
Ek,max  E  W
ER 
e
2
2
Energy of electron level: En 
n2h2
8mL2
Operators
Operator x Eigenfunction = Eigenvalue x
Eigenfunction
Ô  O

Expectation value: O    * Ôdx

Commutator: Ô1, Ô2   Ô1Ô2  Ô2Ô1
X direction: x̂  x
Momentum (x-direction):

P̂x  i
x
2
P̂x 2   2 2
x
Momentum (y-direction):

P̂y  i
y
2
P̂y 2   2 2
y
Momentum (z-direction):

Pz  i
z
2
Pz 2   2 2
z
Angular momentum (x direction):



Lx  i  sin 
 cot  cos  


 
Spin Ŝ  s s  1
Angular momentum (y-direction):



Ly  i   cos 
 cot  sin  


 
Angular momentum (z-direction):

Lz  i

Energy: Ê  
2
2m
2  v r 
Quantum Mechanics
Schrödinger's Equation (TDSE 1D):
2
2 


 V  i
2
2m x
t
i kz   t 
  Ae
Schrödinger's Equation (TISE, 1D):
2
d 2

 V x   E
2m dx 2
dT
i
 ET
dt
  T
Solution for TISE 1D:
1


 n x   
n
 n!2 a  
1
2
x2
 x  2
H n   e 2a
 a  Gaussian
Hermite
Polynomial
Normalization
Exponential
a
m
Schrödinger's Equation (TISE, 2D):
2
  2  2 


 V x, y   E
2m  x 2 y 2 


E  En, x  En, y  nx  ny  1 
Solution for TISE 2D:
x
2
 y2

 x
 y
2
 nx ,ny x, y   H nx   H ny   e 2a
 a
 a
Schrödinger's Equation (TISE, 3D):
2m
 2  r, ,    2 r E  V x  r, ,    0
Normalization of Waveform.
Overall probability = 1

all n, ,
 n,
,m
r,, 
2


* 1
2
all n, ,
Orthonormal waveforms:

1 m  n

*

dx


m
n

0 m  n
1
Equations – Atomic Physics
Linear superposition of two waveforms
  C11  C2  2
 x1, x2  
Normalized if C12  C2 2  1
Represents 2 energy states.
Probability of E1 is C12 , E2 is C2 2
1

Quantized energy: E   n   

2
Quantized energy difference:
 1
1 
E  ER Z 2  2  2 
n 
n
1
Fermions:
 x1 , x2    x2 , x1 
1
 x1, x2  
a x1 b x2  a x2 b x1 
2

Quantized angular momentum:
L2    1 2
n1
Degeneracy of an atom:
2
2
 1  n2
Pauli exclusion principle
No two electrons in the same atom can have
all quantum numbers the same
0
Magnetic Moment of particle in atom
e
L

L  g  M
2m
g and  M depend on the particle being
looked at.
Rotational Energy:
I 2
L2 n n  1
Erot 


2
2m
2
2
n  1
E 
I
1


Spin of a fermion: s 
2
gs
Heisenberg Uncertainty Principle: Et 
Uncertainty: y   y
Identical particles:
2
2
 y

Sizes
Bohr Radius
4 o 2
ao 
me2
ao ' when reduced mass is used
2
Quantized spin: S 2  s s  1
1
a x1 b x2  a x2 b x1 
2
Quantum Numbers
Principle Quantum Number: n  1,2,3,...
Orbital angular momentum quantum number:
 0,1,...,n 1
Azimuthal / Magnetic Quantum Number:
m   ,...,0,...,
1
Spin Quantum Number: ms  
2
j s
2
2
 x1, x2 ,t    A x1  B x2 e
E  EA  EB
Indistinguishable particles:
 iEt
 x1, x2    x2 , x1 
De Broglie Wavelength
h
h
h
 

p mv
2mEk
Compton wavelength if v  c
1

Vibrational energy: Evib   n   

2
Bosons:
 x1 , x2    x2 , x1 
2
2
2