8.04: Quantum Mechanics
Massachusetts Institute of Technology
Professor Allan Adams
Friday May 24 2013
Final Exam Formula Sheet
Fourier Transform Conventions:
∞
1
f (x) = √
2π
dk eikx f˜(k)
−∞
∞
1
f˜(k) = √
2π
dx e−ikx f (x)
−∞
Delta Functions:
∞
dx f (x) δ(x − a) = f (a)
−∞
�
δ(x) =
0
∞
x=0
x=0
∞
1
δ(x) =
2π
�
�
δmn =
dk eikx
−∞
0
1
m=n
m=n
�
Operators and the Schrödinger Equation:
Δx =
x2 ) − (ˆ
(ˆ
x)2
ˆ B]
ˆ = AˆB
ˆ −B
ˆ Aˆ
[A,
p̂ = −ih
∂
∂x
ih
Ê = −
h2 ∂ 2
+ V (x)
2m ∂x2
∂
ψ(x, t) = Eˆ ψ(x, t)
∂t
E φE (x) = −
h2 ∂ 2
φE (x) + V (x) φE (x)
2m ∂x2
Common Integrals:
∞
−x2
dx e
=
√
∞
−∞
−∞
For an infinite square well with 0 ≤ x ≤ L:
�
2
sin (kn x)
φn (x) =
L
kn =
dx f (x)∗ g(x)
(f |g) =
π
(φn |φm ) = δmn
(n + 1)π
L
En =
h2 kn2
2m
Continuity Condition for V (x) = Wo δ(x − a):
φE (a+ ) = φE (a− )
φ/E (a+ ) − φ/E (a− ) =
2mWo
φE (a)
h2
2
8.04: Final Exam Formula Sheet
Physical Constants:
h r 6.6 · 10−16 eV · s
me = 5 · 105 eV /c2
c = 3 · 108 m/s
The Probability Current:
J (x, t) =
ih
2m
ψ
∂ψ ∗
∂ψ
− ψ∗
∂x
∂x
Definition of the S-matrix and the scattering phase:
B
C
A
D
=S
t = |t| e−iϕ ,
,
T = |t|2
Raising and Lowering Operators for the 1d Harmonic Oscillator (β 2 = h/mω):
1
â = √
2
1
β
xˆ + i pˆ
β
h
1
a
ˆ† = √
2
,
1
β
xˆ − i pˆ
β
h
a,
ˆ† = 1
ˆ a
,
Normalization and Orthonormality of 1d HO wavefunctions :
2 /2β 2
φn (x) = An e−x
Hn
x
β
√
An = (2n n! β π)−1/2
(φn |φm ) = δnm
Laplacian in Spherical Coordinates.
p̂V = −ih V
V 2 = 1 ∂ 2r + 1
r r
r2
1 ∂
sin θ ∂θ
sin θ
∂
∂θ
+
1 ∂2
sin2 θ ∂φ2
Angular Momentum Operators in Spherical Coordinates:
L̂z = −ih
∂
,
∂φ
1 ∂
sin θ ∂θ
L̂2 = −h2
sin θ
∂
∂θ
+
1 ∂2
sin2 θ ∂φ2
Angular Momentum Commutators
ˆ z ] = ihL̂x ,
[L̂y , L
ˆ x ] = ihL̂y ,
[L̂z , L
ˆ y ] = ihL̂z ,
[L̂x , L
ˆ 2] = 0
[L̂i , L
Angular Momentum Raising and Lowering Operators
L̂+ = L̂x + iL̂y = he+iφ (i cot θ ∂φ + ∂θ )
[L̂z , L̂± ] = ±h L̂± .
L̂− = L̂x − iL̂y = he−iφ (i cot θ ∂φ − ∂θ )
[L̂2 , L̂± ] = 0 .
First Few Spherical Harmonics
r
r
3
1
Y0,0 =
,
Y1,0 =
cos θ ,
4π
4π
r
Y1,±1 = ∓
3
sin θe±iφ ,
8π
(Ylm |Yl0 m0 ) = δll0 δmm0 .
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8.04 Quantum Physics I
Spring 2013
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