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Review of Quantum Mechanics
2.1 States and Operators
A quantum mechanical system is deﬁned by a Hilbert space, H, whose vectors,
¸
¸
¸ψ
_
are associated with the states of the system. A state of the system is represented by
the set of vectors e
iα
¸
¸
¸ψ
_
. There are linear operators, O
i
which act on this Hilbert
space. These operators correspond to physical observables. Finally, there is an inner
product, which assigns a co

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Review of Quantum Mechanics
2.1 States and Operators
A quantum mechanical system is deﬁned by a Hilbert space,
H
, whose vectors,
ψ
are associated with the states of the system. A state of the system is represented bythe set of vectors
e
iα
ψ
. There are linear operators,
O
i
which act on this Hilbertspace. These operators correspond to physical observables. Finally, there is an innerproduct, which assigns a complex number,
χ
ψ
, to any pair of states,
ψ
,
χ
. Astate vector,
ψ
gives a complete description of a system through the expectationvalues,
ψ
O
i
ψ
(assuming that
ψ
is normalized so that
ψ
ψ
= 1), which wouldbe the average values of the corresponding physical observables if we could measurethem on an inﬁnite collection of identical systems each in the state
ψ
.The adjoint,
O
†
, of an operator is deﬁned according to
χ
O
ψ
=
χ
O
†
ψ
(2.1)In other words, the inner product between
χ
and
O
ψ
is the same as that between
O
†
χ
and
ψ
. An Hermitian operator satisﬁes
O
=
O
†
(2.2)7
Chapter 2: Review of Quantum Mechanics
8while a unitary operator satisﬁes
OO
†
=
O
†
O
= 1 (2.3)If
O
is Hermitian, then
e
i
O
(2.4)is unitary. Given an Hermitian operator,
O
, its eigenstates are orthogonal,
λ
O
λ
=
λ
λ
λ
=
λ
λ
λ
(2.5)For
λ
=
λ
,
λ
λ
= 0 (2.6)If there are
n
states with the same eigenvalue, then, within the subspace spanned bythese states, we can pick a set of
n
mutually orthogonal states. Hence, we can usethe eigenstates
λ
as a basis for Hilbert space. Any state
ψ
can be expanded inthe basis given by the eigenstates of
O
:
ψ
=
λ
c
λ
λ
(2.7)with
c
λ
=
λ
ψ
(2.8)A particularly important operator is the Hamiltonian, or the total energy, whichwe will denote by
H
. Schr¨odinger’s equation tells us that
H
determines how a stateof the system will evolve in time.
i
¯
h∂ ∂t
ψ
=
H
ψ
(2.9)If the Hamiltonian is independent of time, then we can deﬁne energy eigenstates,
H
E
=
E
E
(2.10)
Chapter 2: Review of Quantum Mechanics
9which evolve in time according to:
E
(
t
)
=
e
−
i
Et
¯
h
E
(0)
(2.11)An arbitrary state can be expanded in the basis of energy eigenstates:
ψ
=
i
c
i
E
i
(2.12)It will evolve according to:
ψ
(
t
)
=
j
c
j
e
−
i
Ejt
¯
h
E
j
(2.13)For example, consider a particle in 1
D
. The Hilbert space consists of all continuouscomplex-valued functions,
ψ
(
x
). The position operator, ˆ
x
, and momentum operator,ˆ
p
are deﬁned by:ˆ
x
·
ψ
(
x
)
≡
xψ
(
x
)ˆ
p
·
ψ
(
x
)
≡ −
i
¯
h∂ ∂xψ
(
x
) (2.14)The position eigenfunctions,
xδ
(
x
−
a
) =
aδ
(
x
−
a
) (2.15)are Dirac delta functions, which are not continuous functions, but can be deﬁned asthe limit of continuous functions:
δ
(
x
) = lim
a
→
0
1
a
√
πe
−
x
2
a
2
(2.16)The momentum eigenfunctions are plane waves:
−
i
¯
h∂ ∂xe
ikx
= ¯
hke
ikx
(2.17)Expanding a state in the basis of momentum eigenstates is the same as taking itsFourier transform:
ψ
(
x
) =
∞−∞
dk
˜
ψ
(
k
)1
√
2
πe
ikx
(2.18)
Chapter 2: Review of Quantum Mechanics
10where the Fourier coeﬃcients are given by:˜
ψ
(
k
) =1
√
2
π
∞−∞
dxψ
(
x
)
e
−
ikx
(2.19)If the particle is free,
H
=
−
¯
h
2
2
m∂
2
∂x
2
(2.20)then momentum eigenstates are also energy eigenstates:ˆ
He
ikx
=¯
h
2
k
2
2
me
ikx
(2.21)If a particle is in a Gaussian wavepacket at the srcin at time
t
= 0,
ψ
(
x,
0) =1
a
√
πe
−
x
2
a
2
(2.22)Then, at time
t
, it will be in the state:
ψ
(
x,t
) =1
√
2
π
∞−∞
dka
√
πe
−
i
¯
hk
2
t
2
m
e
−
12
k
2
a
2
e
ikx
(2.23)
2.2 Density and Current
Multiplying the free-particle Schr¨odinger equation by
ψ
∗
,
ψ
∗
i
¯
h∂ ∂tψ
=
−
¯
h
2
2
mψ
∗
∂
2
∇
2
ψ
(2.24)and subtracting the complex conjugate of this equation, we ﬁnd
∂ ∂t
(
ψ
∗
ψ
) =
i
¯
h
2
m
∇·
ψ
∗
∇
ψ
−
∇
ψ
∗
ψ
(2.25)This is in the form of a continuity equation,
∂ρ∂t
=
∇·
j
(2.26)The density and current are given by:
ρ
=
ψ
∗
ψ

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