PHYS 383.3: Formula Sheet for Final Exam
• The time-dependent Schrödinger equation for the wave function Ψ(x, t) is
∂
Ψ(x, t) = ĤΨ(x, t) ,
(1)
∂t
where the Hamiltonian is the sum of the kinetic-energy and potential-energy operators,
ih̄
Ĥ = T̂ + V̂ =
p̂2
+ V (x̂) ,
2m
(2)
with m the mass of the particle and
∂
, x̂ = x
(3)
∂x
in the coordinate representation. The wave function is normalised over the entire system:
Z ∞
dx ρ(x, t) = 1 ,
(4)
p̂ = −ih̄
−∞
where ρ(x, t) = |Ψ(x, t)|2 is the probability density. The probability current is given by
ih̄ ∂Ψ∗ (x, t)
∂Ψ(x, t)
∗
.
(5)
J(x, t) =
Ψ(x, t) − Ψ (x, t)
2m
∂x
∂x
• The expectation value hQi of an operator Q̂ = Q̂(x̂, p̂) is given by
Z ∞
Z ∞
∂
∗
∗
hQi =
dx Ψ (x, t)Q̂(x̂, p̂)Ψ(x, t) =
dx Ψ (x, t)Q̂ x̂, −ih̄
Ψ(x, t)
∂x
−∞
−∞
(6)
in the coordinate representation.
• The standard deviation in the expectation value of operator Q̂ is given by
p
σQ = hQ2 i − hQi2 .
(7)
• The commutator between operators A and B is defined as
[Â, B̂ ] = ÂB̂ − B̂ Â .
(8)
• When the potential is independent of time, the Schrödinger equation in Eq. (1) reduces to
Ĥψn (x) = En ψn (x)
(9)
in the coordinate representation, where n is the quantum number, {En } are the energy
eigenvalues, and {ψn } are the corresponding eigenstates. The eigenstates form a complete,
orthonormal set:
Z ∞
∗
dx ψm
(x)ψn (x) = δnm ,
(10)
−∞
X
ψn (x)ψn∗ (x′ ) = δ(x − x′ ) ,
(11)
n
1
where δnm is the Kronecker delta and δ(x) is the Dirac delta function. An arbitrary function
that satisfies the same boundary condition as the eigenfunctions can be expanded in terms
of them:
Z ∞
X
f (x) =
cn ψn (x) , cn =
dx ψn∗ (x)f (x) .
(12)
−∞
n
• Time evolution of stationary state n with energy En is determined by the exponential
factor, exp(−iEn t/h̄). In general, the time-dependent wave function at time t is given by
X
−iEn t
Ψ(x, t) =
cn exp
ψn (x) ,
(13)
h̄
n
where the expansion coefficients are determined by the initial condition,
Z ∞
dx ψn∗ (x)Ψ(x, 0) .
cn =
(14)
−∞
The expectation value of Ĥ, i.e., the average energy can be calculated either directly by
letting Q̂ = Ĥ in Eq. (6) or by
X
hHi =
|cn |2 En .
(15)
n
• The continuum version of the time-dependent wave function in Eq. (13), e.g., for a free
particle of mass m with no confinement is
Z ∞
1
ih̄k 2 t
Ψ(x, t) = √
dk exp ikx −
φ(k) ,
(16)
2m
2π −∞
where
Z ∞
1
φ(k) = √
dx exp(−ikx)Ψ(x, 0) .
2π −∞
Note that φ(k) and Ψ(x, t) are Fourier transforms of each other.
(17)
• In the Dirac formulation, the eigenvalue equation for a Hermitian operator Q̂ (= Q̂† ) can
be written as
Q̂|ni = qn |ni ,
(18)
where n is the quantum number, {qn } are the eigenvalues, and {|ni} are the corresponding
eigenstates. The orthonormality and completeness of the eigenstates can be expressed as
hm|ni = δnm ,
X
|nihn| = 1 ,
(19)
(20)
n
where 1 is the identity operator. Equations (18), (19) and (20) correspond to Eqs. (9), (10)
and (11), respectively, for Q̂ = Ĥ. Equation (12) can be expressed in the bracket notation
as
X
X
|f i = 1|f i =
|nihn|f i =
cn |ni , cn = hn|f i ,
(21)
n
n
2
where |f i is an arbitrary state in the Hilbert space spanned by {|ni} as the basis vectors.
The continuum version of Eq. (20), e.g., for the eigenstates of p̂ and x̂ is, respectively,
Z ∞
Z ∞
dx |xihx| = 1 .
(22)
dp |pihp| = 1 ,
−∞
−∞
• The commutator between the position and coordinate operators is always given by
[x̂, p̂ ] = ih̄ ,
(23)
∂
,
∂p
(24)
while the operators are
x̂ = ih̄
p̂ = p
in the momentum representation.
• The generalised uncertainty relation between the standard deviations σA and σB , respectively, for operators  and B̂ is given by
σA σB ≥
1
h [Â, B̂ ] i ,
2i
(25)
which reduces to σx σp ≥ h̄/2 for x̂ and p̂.
• Time evolution of the expectation value hQi is determined by
i
∂ Q̂
dhQi
+ h [Ĥ, Q̂ ] i .
=
dt
∂t
h̄
(26)
• The time-dependent Schrödinger equation for an arbitrary state |S(t)i at time t can be
written in the Dirac formulation as
ih̄
∂
|S(t)i = Ĥ|S(t)i ,
∂t
which reduces to Eq. (1) for Ψ(x, t) ≡ hx|S(t)i. The solution to Eq. (27) is
j
∞
X
1 −iĤt
−iĤt
|S(0)i =
|S(0)i .
|S(t)i = exp
h̄
j!
h̄
j=0
(27)
(28)
One can expand the initial state |S(0)i in terms of the eigenstates {|ni} of the Hamiltonian
with the eigenvalues {En } (Q̂ = Ĥ and qn = En in Eq. (18)),
X
|S(0)i =
cn |ni , cn = hn|S(0)i ,
(29)
n
which is equivalent to Eq. (21) with |f i = |S(0)i. Corresponding to Eqs. (13) and (14), at
later time t,
X
−iĤt
−iEn t
|S(t)i = exp
|S(0)i =
cn exp
|ni .
(30)
h̄
h̄
n
3
• The eigenvalues and eigenstates of a particle of mass m in a one-dimensional infinite square
well of width a are given by
r
h̄2 π 2 2
nπx
2
n , ψn (x) =
sin
En =
; n = 1, 2, . . . .
(31)
2ma2
a
a
• The raising and lowering operators for a particle of mass m in the harmonic oscillator
potential with frequency ω, V (x) = 21 mω 2 x2 , are defined by
â± = √
1
(∓ip̂ + mωx̂) .
2h̄mω
(32)
Namely,
x̂ =
and
r
r
h̄
h̄mω
(â+ + â− ) , p̂ = i
(â+ − â− ) ,
2mω
2
1
.
Ĥ = h̄ω â+ â− +
2
Denoting the complete orthonormal set of eigenstates of Ĥ as {|ni; n = 0, 1, 2, . . . },
√
√
â+ |ni = n + 1|n + 1i , â− |ni = n|n − 1i .
(33)
(34)
(35)
Note that â− |0i = 0.
• Simultaneous eigenstates of J 2 and Jz , where the angular momentum operator J can be
the orbital (J = L), spin (J = S), or total angular momentum (J = L + S):
J 2 |j, mi = j(j + 1)h̄2 |j, mi ,
Jz |j, mi = mh̄ |j, mi ; m = −j, −j + 1, . . . , j − 1, j .
(36)
(37)
When J = L, j = 0, 1, 2, . . . and when J = S, j can be either integer (j = 0, 1, 2, . . . ) or
half-integer (j = 1/2, 3/2, . . . ).
• Addition of two angular momenta: for J = J1 + J2 , where
Ji2 |ji , mi i = ji (ji + 1)h̄2 |ji , mi i ,
Jzi |ji , mi i = mi h̄ |ji , mi i ; mi = −ji , −ji + 1, . . . , ji − 1, ji
(38)
(39)
for i = 1, 2, the simultaneous eigenstates of the total angular momentum are
J 2 |j, mi = j(j + 1)h̄2 |j, mi , j = |j1 − j2 |, |j1 − j2 | + 1, . . . , j1 + j2 − 1, j1 + j2 ,(40)
Jz |j, mi = mh̄ |j, mi , m = −j, −j + 1, . . . , j − 1, j .
(41)
Namely, j takes on all values between |j1 − j2 | and j1 + j2 in steps of unity.
• For s = 1/2 (spin 1/2), the simultaneous eigenstates of S 2 and Sz are
3
S 2 |±i = h̄2 |±i ,
4
1
Sz |±i = ± h̄|±i ,
2
4
(42)
(43)
where
1
1
|±i = , ±
2
2
(44)
are called the “up” (|+i ≡ | ↑i) and “down” (|−i ≡ | ↓i) states. The matrix representation
of the spin angular momentum can be written as S = h̄2 σ, where the x, y, and z components
of σ are the Pauli matrices:
1 0
0 −i
0 1
.
(45)
,
σz =
,
σy =
σx =
0 −1
i 0
1 0
• Perturbation theory: in case there is no degeneracy in the unperturbed states, the firstorder and second-order correction due to perturbation Ĥ ′ to the energy of state n are given
by, respectively,
En(1) = hn|Ĥ ′ |ni ,
X |hm|Ĥ ′ |ni|2
,
En(2) =
(0)
(0)
m6=n En − Em
(46)
(47)
where {|ni} are the eigenstates of the unperturbed Hamiltonian Ĥ (0) :
Ĥ (0) |ni = En(0) |ni .
(48)
In case there is degeneracy in the unperturbed energy level, diagonalise the matrix representation of Ĥ ′ in the subspace of degenerate states: the eigenvalues will be the first-order
energy correction.
5