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Physics 195a
Course Notes
Ideas of Quantum Mechanics
021024 F. Porter
1 Introduction
This note summarizes and examines the foundations of quantum mechanics,
including the mathematical background.
2 General Review of the Ideas of Quantum
Mechanics
2.1
States
We have in mind that there is a “system”, which is describable in terms
of possible “states”. A system could be something simple, such as a single
electron, or complex, such as a table.
Suppose we have a system consisting of
N
spinless particles. We use
the term “particle” to denote any object for which any internal structure
is unimportant. Classically, we may describe the state of this system by
specifying, at some time
t
the generalized coordinates and momenta:
{
q
i
(
t
)
,p
i
(
t
)
,i
=1
,
2
,...,N
}
,
(1)
where the spatial dimensionality of the
q
i
and
p
i
is implicit. The time evolu
tion of this system is given by Hamilton’s equations:
˙
q
i
=
∂H
∂p
i
(2)
−
˙
p
i
=
∂q
i
(3)
In quantum mechanics, it is not possible to give such a complete speci
Fcation to arbitrary precision. ±or example, the limit to how well we may
specify the position and momentum of a particle in one dimension is limited
by the “uncertainty principle”: ∆
x
∆
p
≥
1
/
2, where ∆ indicates a range of
possible values. We’ll investigate this relation more explicitly later, but for
1
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View Full Document now it should just be a reminder of your elementary quantum mechanics un
derstanding. We must be content with selecting a suitable set of quantities
which can be simultaneously speciFed to describe the state. We refer to this
set as a “Complete Set of Commuting Observables” (CSCO).
Specifying a CSCO corresponds to specifying the eigenvalues of an ap
propriate complete set of commuting Hermitian operators, for the state in
question. Measurements (eigenvalues of Hermitian operators) of other quan
tities cannot be predicted with certainty, only probabilities of outcomes can
be given. The evolution in time of the system is described by a “wave equa
tion”, for example, the Schr¨
odinger equation.
2.2
Probability Amplitudes
The quantum mechanical state of a system is described in terms of waves,
called
probability amplitudes
, or just “amplitudes” for short. Note that
probabilities themselves are always nonnegative, so it is more diﬃcult to
imagine the probabilities themselves as wavelike. Instead, the probabilities
are obtained by squaring the amplitudes:
Probability
∼
ψ

2
,
(4)
where
ψ
stands for the amplitude. More explicitly, the probability of ob
serving state variable (
e.g.
, position)
x
in volume element
d
3
(
x
) around
x
is
equal to:

ψ
(
x
)

2
d
3
(
x
)
.
(5)
A quantum mechanical probability is analogous to the intensity of a classical
wave.
The quantum mechanical wave evolves in time according to a time evo
lution operator,
e
−
iHt
involving the Hamiltonian,
H
. Hence, if
e
−
iHt
ψ
0
(
x
)=
ψ
(
x
,t
)
,
(6)
where
ψ
0
(
x
) is the wave function at
t
= 0 in terms of coordinate position
x
,
then di±erentiation gives:
i
dψ
(
x
)
dt
=
Hψ
(
x
)
.
(7)
We recognize this as the Schr¨odinger equation. Thus, the temporal frequency
of the wave is determined by the energy structure. ²or a particle of energy
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This document was uploaded on 03/21/2012.
 Winter '09
 Physics, mechanics

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