Physics 6210/Spring 2007/Lecture 18
Lecture 18
Relevant sections in text:
§
2.3
Stationary states and classical mechanics
Here we use the oscillator to illustrate a very key point about the relation between
classical and quantum mechanics.
The stationary states we just studied do not provide any explicitly dynamical behavior.
This is not a speciﬁc feature of the oscillator, but a general feature of stationary states in
quantum mechanics. This is, at ﬁrst sight, a little weird compared to classical mechanics.
Think about it: the probability distributions for position and momentum are time inde
pendent in any state of deﬁnite energy. In classical mechanics the position and momentum
(and the energy) can, at each moment of time, be determined with certainty; the values
for the position and momentum oscillate in time in every state but the ground state.*
In light of the incompatibility of the position, momentum and energy observables in the
quantum description, one cannot directly compare the classical and quantum predictions
in the excited states. The quantum predictions are purely statistical, involving repeated
state preparations – states speciﬁed only by their energy – and measurements of various
observables. If we want to compare the quantum and classical descriptions we need to ask
the right questions of classical mechanics — this means statistical questions. Let us pause
for a moment to expand on this.
In classical mechanics every solution of the equations of motion is a state of deﬁnite
energy (assuming energy is conserved). Fix an energy
E
. Let us ask the same question
that we ask in quantum theory: Given the energy what is the probability for ﬁnding
the particle at various locations? To answer this question we deﬁne the probability for
ﬁnding the classical particle to be in the interval [
x, x
+
dx
] to be proportional to the time
spent in that region. (The proportionality constant is used to normalize the probability
distribution.) For a classical oscillator at the point
x
, a displacement
dx
takes place in the
time
dt
where (exercise)
dt
=
dx
q
2
E
m

ω
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 Spring '07
 M
 mechanics, Probability, Probability distribution

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