Formulas to Memorize for Exam 1
revised October 3, 2011
The following is a list of formulas that you need to know for Exam 1. I suspect that you
already know most of them from working the problems. The first part of the exam will be
short answer questions involving these formulas.
1. General
(a) Time dependent Schrodinger equation:
h̄2 ∂ 2 ψ
∂ψ
=−
+Vψ
ih̄
∂t
2m ∂x2
(b) Expectations values:
Z
hxi =
dx ψ ∗ (x, t) x ψ(x, t)
Z
hpi =
dx ψ ∗ (x, t) p ψ(x, t),
where p = −ih̄∂/∂x. Other expectation values, e.g. hx2 i and hp2 i, may be computed by substituting the appropriate expectation values instead of x and p above.
(c) Uncertainty principle:
σx =
q
hx2 i − hxi2
q
hp2 i − hpi2
σp =
σx σp ≥ h̄/2.
(d) Time independent Schrodinger equation:
Eϕ(x) = −
h̄2 d2 ϕ
+ V ϕ = Hϕ
2m dx2
(e) Completeness:
If Hϕn = En ϕn are the solutions to the time indpendent Schrodinger equation,
then the solution to the time dependent Schrodinger equation have the form
ψ(x, t) =
X
cn ϕn (x)e−iEn t/h̄ ,
n
where
cn =
Z
dx ϕ∗n (x) ψ(x, 0).
(f) Orhonormality:
For different energies the wave functions are orthogonal. The wave functions are
also normalized.
Z
dx ϕ∗m (x) ϕn (x) = δm,n
2. Infinite square well
s
2
sin(kx)
a
nπ
k =
a
h̄2 k 2
En =
, for n = 1, 2, 3, . . .
2m
ϕn (x) =
3. Harmonic oscillator
H =
En =
[a, a+ ]
[x, p]
a+ ψn
a− ψn
a+ a− ψn
4. Free particle
=
=
=
=
=
1
h̄ω a+ a− +
2
1
, for n = 0, 1, 2, . . .
h̄ω n +
2
1
ih̄
√
n + 1 ψn+1
√
n ψn−1
n ψn
∞
h̄k2
1
φ(k) ei(kx− 2m t) dk
ψ(x, t) = √
2π −∞
Z ∞
1
φ(k) = √
ψ(x, 0) e−ikx dx
2π −∞
Z
5. Piecewise constant potentials
The solution to the time independent Schrodinger equation for piecewise constant
potentials (constant Vo ),
h̄2 d2 ϕ
+ Vo ϕ,
Eϕ(x) = −
2m dx2
is
s
2m(E − Vo )
ikx
′ −ikx
ϕ(x) = Ae + A e
with k =
for E > Vo ,
h̄2
and
s
2m(Vo − E)
ρx
′ −ρx
ϕ(x) = Be + B e
with ρ =
for E < Vo .
h̄2
The wave function and its first derivative are continuous at the boundaries (except for
delta function potentials). The probability current is
!
∂ψ
∂ψ ∗
h̄
ψ∗
.
−ψ
j=
2mi
∂x
∂x
It satisfies the continuity equation for the probabiltiy
∂|ψ(x, t)|2 ∂j
+
= 0,
∂t
∂x
and for stationary states (solutions to the time independent Schrodinger equation)
satisfies ∂j/∂x = 0. The transmission and reflection probabilities are
jtransmitted
jincoming
jref lected
R =
jincoming
T + R = 1.
T =
Formulas Printed on Exam 1
These formulas will be printed on the exam. You do not need to memorize them.
Harmonic oscillator:
1
(−ip + mωx)
2h̄mω
1
√
(+ip + mωx)
2h̄mω
s
h̄
(a+ + a− )
2mω
s
h̄mω
i
(a+ − a− )
2
mω 2
mω 1/4
x
exp −
πh̄
2h̄
1
√ (a+ )n ψ0
n!
a+ = √
a− =
x =
p =
ψ0 (x) =
ψn =
Delta function potential V (x) = α δ(x):
dϕ(0+ )
dϕ(0− )
2mα
−
= 2 ϕ(0).
dx
dx
h̄