Quantum mechanics review
description
Transcript of Quantum mechanics review
![Page 1: Quantum mechanics review](https://reader035.fdocuments.us/reader035/viewer/2022062422/56813ff2550346895dab0ab3/html5/thumbnails/1.jpg)
Quantum mechanics review
![Page 2: Quantum mechanics review](https://reader035.fdocuments.us/reader035/viewer/2022062422/56813ff2550346895dab0ab3/html5/thumbnails/2.jpg)
• 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
![Page 3: Quantum mechanics review](https://reader035.fdocuments.us/reader035/viewer/2022062422/56813ff2550346895dab0ab3/html5/thumbnails/3.jpg)
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
![Page 4: Quantum mechanics review](https://reader035.fdocuments.us/reader035/viewer/2022062422/56813ff2550346895dab0ab3/html5/thumbnails/4.jpg)
Physical properties: Expectation values
nAnA
or
dAA nn
*
Dirac notation or bra-ket notation
![Page 5: Quantum mechanics review](https://reader035.fdocuments.us/reader035/viewer/2022062422/56813ff2550346895dab0ab3/html5/thumbnails/5.jpg)
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
![Page 6: Quantum mechanics review](https://reader035.fdocuments.us/reader035/viewer/2022062422/56813ff2550346895dab0ab3/html5/thumbnails/6.jpg)
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
![Page 7: Quantum mechanics review](https://reader035.fdocuments.us/reader035/viewer/2022062422/56813ff2550346895dab0ab3/html5/thumbnails/7.jpg)
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
![Page 8: Quantum mechanics review](https://reader035.fdocuments.us/reader035/viewer/2022062422/56813ff2550346895dab0ab3/html5/thumbnails/8.jpg)
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
![Page 9: Quantum mechanics review](https://reader035.fdocuments.us/reader035/viewer/2022062422/56813ff2550346895dab0ab3/html5/thumbnails/9.jpg)
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
![Page 10: Quantum mechanics review](https://reader035.fdocuments.us/reader035/viewer/2022062422/56813ff2550346895dab0ab3/html5/thumbnails/10.jpg)
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
![Page 11: Quantum mechanics review](https://reader035.fdocuments.us/reader035/viewer/2022062422/56813ff2550346895dab0ab3/html5/thumbnails/11.jpg)
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.
![Page 12: Quantum mechanics review](https://reader035.fdocuments.us/reader035/viewer/2022062422/56813ff2550346895dab0ab3/html5/thumbnails/12.jpg)
![Page 13: Quantum mechanics review](https://reader035.fdocuments.us/reader035/viewer/2022062422/56813ff2550346895dab0ab3/html5/thumbnails/13.jpg)
Harmonic Oscillator 1-d
F=-k(x-x0) Internal coordinates; Set x0=0
22
22
21;2 kxVV
dx
dH
![Page 14: Quantum mechanics review](https://reader035.fdocuments.us/reader035/viewer/2022062422/56813ff2550346895dab0ab3/html5/thumbnails/14.jpg)
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
![Page 15: Quantum mechanics review](https://reader035.fdocuments.us/reader035/viewer/2022062422/56813ff2550346895dab0ab3/html5/thumbnails/15.jpg)
25.011 2 xexN
kE )(v 21
v
25.000
xeN
![Page 16: Quantum mechanics review](https://reader035.fdocuments.us/reader035/viewer/2022062422/56813ff2550346895dab0ab3/html5/thumbnails/16.jpg)
http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc5.html#c1
![Page 17: Quantum mechanics review](https://reader035.fdocuments.us/reader035/viewer/2022062422/56813ff2550346895dab0ab3/html5/thumbnails/17.jpg)
![Page 18: Quantum mechanics review](https://reader035.fdocuments.us/reader035/viewer/2022062422/56813ff2550346895dab0ab3/html5/thumbnails/18.jpg)
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
![Page 19: Quantum mechanics review](https://reader035.fdocuments.us/reader035/viewer/2022062422/56813ff2550346895dab0ab3/html5/thumbnails/19.jpg)
Rotation: Rigid Rotor
01
22
2
VprLRII
LH
0,2
2222
i
zyx
LL
LLLL
![Page 20: Quantum mechanics review](https://reader035.fdocuments.us/reader035/viewer/2022062422/56813ff2550346895dab0ab3/html5/thumbnails/20.jpg)
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
![Page 21: Quantum mechanics review](https://reader035.fdocuments.us/reader035/viewer/2022062422/56813ff2550346895dab0ab3/html5/thumbnails/21.jpg)
Degeneracy
![Page 22: Quantum mechanics review](https://reader035.fdocuments.us/reader035/viewer/2022062422/56813ff2550346895dab0ab3/html5/thumbnails/22.jpg)
Angular Momemtum operators the spherical harmonics
Operators L2 ansd Lz
llmm
llmmz
lmlm
lmlmz
lllmLml
mlmLml
YllYL
YmYL
''2
''
2
)1(''
''
)1(
![Page 23: Quantum mechanics review](https://reader035.fdocuments.us/reader035/viewer/2022062422/56813ff2550346895dab0ab3/html5/thumbnails/23.jpg)
Rotation: Rigid Rotor
01
22
2
VprLRII
LH
Eigenvalues are thus:
llmmI
lllm
I
Lml ''
2
2
)1(
2''
l = 0,1,2,3,…
![Page 24: Quantum mechanics review](https://reader035.fdocuments.us/reader035/viewer/2022062422/56813ff2550346895dab0ab3/html5/thumbnails/24.jpg)
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.
![Page 25: Quantum mechanics review](https://reader035.fdocuments.us/reader035/viewer/2022062422/56813ff2550346895dab0ab3/html5/thumbnails/25.jpg)
The hydrogen atom
r
emH
222
2
Set up problem in spherical polar coordinates. Hamiltonian is separable into radial and angular components
,lmnlnlm YrR
![Page 26: Quantum mechanics review](https://reader035.fdocuments.us/reader035/viewer/2022062422/56813ff2550346895dab0ab3/html5/thumbnails/26.jpg)
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
![Page 27: Quantum mechanics review](https://reader035.fdocuments.us/reader035/viewer/2022062422/56813ff2550346895dab0ab3/html5/thumbnails/27.jpg)
molekJeVRn
R
n
eEn /13126.13;
2 222
4
![Page 28: Quantum mechanics review](https://reader035.fdocuments.us/reader035/viewer/2022062422/56813ff2550346895dab0ab3/html5/thumbnails/28.jpg)
![Page 29: Quantum mechanics review](https://reader035.fdocuments.us/reader035/viewer/2022062422/56813ff2550346895dab0ab3/html5/thumbnails/29.jpg)
![Page 30: Quantum mechanics review](https://reader035.fdocuments.us/reader035/viewer/2022062422/56813ff2550346895dab0ab3/html5/thumbnails/30.jpg)
![Page 31: Quantum mechanics review](https://reader035.fdocuments.us/reader035/viewer/2022062422/56813ff2550346895dab0ab3/html5/thumbnails/31.jpg)
![Page 32: Quantum mechanics review](https://reader035.fdocuments.us/reader035/viewer/2022062422/56813ff2550346895dab0ab3/html5/thumbnails/32.jpg)
![Page 33: Quantum mechanics review](https://reader035.fdocuments.us/reader035/viewer/2022062422/56813ff2550346895dab0ab3/html5/thumbnails/33.jpg)
![Page 34: Quantum mechanics review](https://reader035.fdocuments.us/reader035/viewer/2022062422/56813ff2550346895dab0ab3/html5/thumbnails/34.jpg)
![Page 35: Quantum mechanics review](https://reader035.fdocuments.us/reader035/viewer/2022062422/56813ff2550346895dab0ab3/html5/thumbnails/35.jpg)
![Page 36: Quantum mechanics review](https://reader035.fdocuments.us/reader035/viewer/2022062422/56813ff2550346895dab0ab3/html5/thumbnails/36.jpg)
![Page 37: Quantum mechanics review](https://reader035.fdocuments.us/reader035/viewer/2022062422/56813ff2550346895dab0ab3/html5/thumbnails/37.jpg)
![Page 38: Quantum mechanics review](https://reader035.fdocuments.us/reader035/viewer/2022062422/56813ff2550346895dab0ab3/html5/thumbnails/38.jpg)
![Page 39: Quantum mechanics review](https://reader035.fdocuments.us/reader035/viewer/2022062422/56813ff2550346895dab0ab3/html5/thumbnails/39.jpg)
![Page 40: Quantum mechanics review](https://reader035.fdocuments.us/reader035/viewer/2022062422/56813ff2550346895dab0ab3/html5/thumbnails/40.jpg)
![Page 41: Quantum mechanics review](https://reader035.fdocuments.us/reader035/viewer/2022062422/56813ff2550346895dab0ab3/html5/thumbnails/41.jpg)
![Page 42: Quantum mechanics review](https://reader035.fdocuments.us/reader035/viewer/2022062422/56813ff2550346895dab0ab3/html5/thumbnails/42.jpg)
![Page 43: Quantum mechanics review](https://reader035.fdocuments.us/reader035/viewer/2022062422/56813ff2550346895dab0ab3/html5/thumbnails/43.jpg)
![Page 44: Quantum mechanics review](https://reader035.fdocuments.us/reader035/viewer/2022062422/56813ff2550346895dab0ab3/html5/thumbnails/44.jpg)