Quantum Mechanics
Contents
1 The Wave Function 2
2 Formalism 5
3 Quantum Mechanics in Three Dimensions 6
4 Identical Particles 8
5 Atomic Structure 8
6 Perturbation Theory 9
1
,1 The Wave Function
Wave Function : The state of a quantum mechanical system is described
by its wave function Ψ(x, t). The wave function is found by solving the
Schrödinger Equation :
∂Ψ(x, t) ~2 ∂ 2 Ψ(x, t)
i~ =− + V (x, t)Ψ(x, t)
∂t 2m ∂x2
Ra
Born’s statistical interpretation is that b |Ψ(x0 , t0 )|2 dx gives the probabil-
ity of finding the particle between positions a and b at time t (the integral
must be normalized). When measurements provide a position c, the wave
function is said to have collapsed to a spike at c.
Operators : An operator is a function that represents a physical state of
∂
a system. For position this is x̂ = x and for momentum p̂ = −i~ ∂x . All
operators of classical variables Q̂ can be expressed as functions of p̂ and x̂,
p̂2
for instance T̂ = 2m . The expectation value is:
Z
hQ̂(x, p)i = Ψ(x, t)Q̂(x, p)Ψ(x, t) dx
Ehrenfest’s Theorem : Expectation values obey classical laws, so
dhpi ∂V (x, t) dhxi
= − and hvi =
dt ∂x dt
De Broglie Formula : The wavelength of a particle is related to it’s mo-
mentum by:
2π~
p=
λ
Heisenberg’s Uncertainty Principle : The more precise a wave’s position
is, the less precise its wavelength. This observation translates to:
~
σx σp ≥
2
where σx and σp are the standard deviation is x and p respectively.
Hamiltonian : The total energy of the system is described by the Hamilto-
nian:
p2 ~2 ∂ 2
H(x, p) = + V (x, t), Ĥ(x, p) = − + V (x, t)
2m 2m ∂x2
2
, Time-independent Schrödinger Equation : Assume that V is indepen-
dent of t. Then the time-independent Schrödinger equation is:
~2 d2 ψ(x)
− + V (x)ψ(x) = Eψ(x) ⇔ Ĥψ = Eψ
2m dx2
Where ψ is a function of x only. For separable solutions, that is Ψ(x, t) =
ϕ(t)ψ(x), the general solution is a linear combination of separable solutions.
So we have a general solution:
∞
X iEn t
Ψ(x, t) = cn ψn (x)e− ~
n=1
for each allowed energy En . Separable solutions are stationary states, mean-
ing that every expectation value is constant in time.
Infinite Square Well : Suppose V (x) = 0 if 0 ≤ x ≤ a and V (x) = ∞ oth-
erwise. The time-independent Schrödinger equation is the simple harmonic
oscillator equation:
√
d2 ψ 2 2mE
2
= −k ψ, where k :=
dx ~
n2 π 2 ~2
The possible values of E are En = 2ma2
and has solutions:
r
2 nπx
ψn = sin
a a
R
The ψ’s are orthonormal, that is ψm (x)ψn (x) dx = δnm , and they are
complete. ThisPmeans that any function f (x) can be expressed as its Fourier
series; f (x) = ∞
n=1 cn ψn (x). So the nth coefficient of the expansion is:
Z ∞
cn = ψn (x)f (x) dx
−∞
|cn |2 give the probability of finding an energy of En . This probability is
independent of time, which makes it a manifestation of conservation of energy
in Quantum Mechanics. The expectation value of the Hamiltonian is:
∞
X
hHi = |cn |2 En
n=1
3
Contents
1 The Wave Function 2
2 Formalism 5
3 Quantum Mechanics in Three Dimensions 6
4 Identical Particles 8
5 Atomic Structure 8
6 Perturbation Theory 9
1
,1 The Wave Function
Wave Function : The state of a quantum mechanical system is described
by its wave function Ψ(x, t). The wave function is found by solving the
Schrödinger Equation :
∂Ψ(x, t) ~2 ∂ 2 Ψ(x, t)
i~ =− + V (x, t)Ψ(x, t)
∂t 2m ∂x2
Ra
Born’s statistical interpretation is that b |Ψ(x0 , t0 )|2 dx gives the probabil-
ity of finding the particle between positions a and b at time t (the integral
must be normalized). When measurements provide a position c, the wave
function is said to have collapsed to a spike at c.
Operators : An operator is a function that represents a physical state of
∂
a system. For position this is x̂ = x and for momentum p̂ = −i~ ∂x . All
operators of classical variables Q̂ can be expressed as functions of p̂ and x̂,
p̂2
for instance T̂ = 2m . The expectation value is:
Z
hQ̂(x, p)i = Ψ(x, t)Q̂(x, p)Ψ(x, t) dx
Ehrenfest’s Theorem : Expectation values obey classical laws, so
dhpi ∂V (x, t) dhxi
= − and hvi =
dt ∂x dt
De Broglie Formula : The wavelength of a particle is related to it’s mo-
mentum by:
2π~
p=
λ
Heisenberg’s Uncertainty Principle : The more precise a wave’s position
is, the less precise its wavelength. This observation translates to:
~
σx σp ≥
2
where σx and σp are the standard deviation is x and p respectively.
Hamiltonian : The total energy of the system is described by the Hamilto-
nian:
p2 ~2 ∂ 2
H(x, p) = + V (x, t), Ĥ(x, p) = − + V (x, t)
2m 2m ∂x2
2
, Time-independent Schrödinger Equation : Assume that V is indepen-
dent of t. Then the time-independent Schrödinger equation is:
~2 d2 ψ(x)
− + V (x)ψ(x) = Eψ(x) ⇔ Ĥψ = Eψ
2m dx2
Where ψ is a function of x only. For separable solutions, that is Ψ(x, t) =
ϕ(t)ψ(x), the general solution is a linear combination of separable solutions.
So we have a general solution:
∞
X iEn t
Ψ(x, t) = cn ψn (x)e− ~
n=1
for each allowed energy En . Separable solutions are stationary states, mean-
ing that every expectation value is constant in time.
Infinite Square Well : Suppose V (x) = 0 if 0 ≤ x ≤ a and V (x) = ∞ oth-
erwise. The time-independent Schrödinger equation is the simple harmonic
oscillator equation:
√
d2 ψ 2 2mE
2
= −k ψ, where k :=
dx ~
n2 π 2 ~2
The possible values of E are En = 2ma2
and has solutions:
r
2 nπx
ψn = sin
a a
R
The ψ’s are orthonormal, that is ψm (x)ψn (x) dx = δnm , and they are
complete. ThisPmeans that any function f (x) can be expressed as its Fourier
series; f (x) = ∞
n=1 cn ψn (x). So the nth coefficient of the expansion is:
Z ∞
cn = ψn (x)f (x) dx
−∞
|cn |2 give the probability of finding an energy of En . This probability is
independent of time, which makes it a manifestation of conservation of energy
in Quantum Mechanics. The expectation value of the Hamiltonian is:
∞
X
hHi = |cn |2 En
n=1
3
