quantum mechanics review. reading for week of 1/28-2/1 – chapters 1, 2, and 3.1,3.2 reading for...
TRANSCRIPT
Quantum mechanics review
• Reading for week of 1/28-2/1– Chapters 1, 2, and 3.1,3.2
• Reading for week of 2/4-2/8– Chapter 4
Schrodinger Equation (Time-independent)
where
The solutions incorporate boundary conditions and are a family of eigenvalues with increasing energy and corresponding eigenvectors with an increasing number of nodes.
The solutions are orthonormal.
nn EH
nmmn d *
VmH
VTH
22
2
Physical properties: Expectation values
nAnA
or
dAA nn
*
Dirac notation or bra-ket notation
Physical properties: Hermitian Operators
mAnAnAmAnmmn
Real Physical Properties are Associated with Hermitian Operators
Hermitian operators obey the following:
The value <A>mn is also known as a matrix element, associated with solving the problem of the expectation value for A as the eigenvalues of a matrix indexed by m and n
Zero order models:
Particle-in-a-box: atoms, bonds, conjugated alkenes, nano-particles
Harmonic oscillator: vibrations of atoms
Rigid-Rotor: molecular rotation; internal rotation of methyl groups, motion within van der waals molecules
Hydrogen atom: electronic structure
Hydrogenic Radial Wavefunctions
Particle-in-a-3d-Box
x
a
V(x)
V(x) =0; 0<x<a
V(x) =∞; x>a; x <0
b y ; c z
c
zn
b
yn
a
xn
abczyx
nnn zyx
sinsinsin
8nx,y,z = 1,2,3, ...
V
zyxmVmVTH2
2
2
2
2
2222
22
Particle-in-a-3d-Box
x
a
V(x)
V(x) =0; 0<x<a
V(x) =∞; x>a; x <0
b y ; c z
2
2
2
2
2
22
8 c
n
b
n
a
n
m
hE zyx
nnn zyx
0111
8 2
2
2
2
2
22
111
cbam
hE zyx
Zero point energy/Uncertainty Principle
In this case since V=0 inside the box E = K.E.
If E = 0 the p = 0 , which would violate the uncertainty principle.
2
px
Zero point energy/Uncertainty Principle
More generally
Variance or rms:
If the system is an eigenfunction of then is precisely determined and there is no variance.
A
A
22
AAA
2,
pxixppxpx
Zero point energy/Uncertainty Principle
BABA ,2
1
If the commutator is non-zero then the two properties cannot be precisely defined simultaneously. If it is zero they can be.
Harmonic Oscillator 1-d
F=-k(x-x0) Internal coordinates; Set x0=0
22
22
21;2 kxVV
dx
dH
Hermite polynomials
535
424
33
22
1
0
v
3216120
164812
812
42
2
1
)1(22
v
qqqqH
qqqH
qqqH
qqH
qqH
qH
edq
deqH q
n
nq
Harmonic Oscillator Wavefunctions
2
1
vv !v2
N
V = quantum number = 0,1,2,3
/
Hv = Hermite polynomials Nv = Normalization Constant
25.0vvv
xeHN x
25.011 2 xexN
kE )(v 21
v
25.000
xeN
http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc5.html#c1
Raising and lowering operators:
Recursion relations used to define new members in a family of solutions to D.E.
lowering2
raising2
0
0
0
0
ˆ
ˆ
piX
piX
a
a
1
11
VVVa
VVVa
Rotation: Rigid Rotor
01
22
2
VprLRII
LH
0,2
2222
i
zyx
LL
LLLL
Rotation: Rigid Rotor
Wavefunctions are the spherical harmonics
imml
mm
lm
ePml
mll
Y
)(cos!
!
4
12)1(
,
lmlm
lmlmz
YllYL
YmYL
)1(2
Operators L2 ansd Lz
,lmlm Y
Degeneracy
Angular Momemtum operators the spherical harmonics
Operators L2 ansd Lz
llmm
llmmz
lmlm
lmlmz
lllmLml
mlmLml
YllYL
YmYL
''2
''
2
)1(''
''
)1(
Rotation: Rigid Rotor
01
22
2
VprLRII
LH
Eigenvalues are thus:
llmmI
lllm
I
Lml ''
2
2
)1(
2''
l = 0,1,2,3,…
Lots of quantum mechanical and spectroscopic problems have solutions that can be usefully expressed as sums of spherical harmonics.
e.g. coupling of two or more angular momentumplane wavesreciprocal distance between two points in space
Also many operators can be expressed as spherical harmonics:
lmYml LM''The properties of the matrix element above are well known and are zero unless
-m’+M+m = 0l’+L+l is even
Can define raising and lowering operators for these wavefunctions too.
The hydrogen atom
r
emH
222
2
Set up problem in spherical polar coordinates. Hamiltonian is separable into radial and angular components
,lmnlnlm YrR
n
the principal quantum number, determines energy
l
the orbital angular momentum quantum number
l= n-1, n-2, …,0
m
the magnetic quantum number -l, -l+1, …, +l
molekJeVRn
R
n
eEn /13126.13;
2 222
4
