Download - PHYSICS 201b Quantum notes R. Shankar 2010
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For these values of m each exponential function is periodic with period L. Second, to find
the coefficients Am in Eq. (155), we can either get the answer by inspection if ψ is a sum of
trigonometric functions, or in every case, turn to
Am =
� L
0
1√L
e−2πimx/L
ψ(x)dx. (157)
Third, if we first normalize ψ(x) or it is already normalized,
P (m) = |Am|2 (158)
is the absolute probability. That is�
m |Am|2 = 1 and there is no need to divide by it.
Postulate 5 If x is measured, ψ will collapse to a spike at the measured value of x and
if p is measured ψ will collapse to the one term in the sum Eq. (155) corresponding to the
measured value.
Postulate 6 A particle in a state of definite energy E obeys the time-independent
Schrödinger Equation
− �2
2m
d2ψE(x)
dx2+ V (x)ψE(x) = EψE(x).
As with momentum any ψ(x) can be written as
ψ(x) =�
E
AEψE(x)
where the coefficients are
AE =
�ψ∗E(x)ψ(x)dx
provided ψE(x) is normalized).
Again, if energy is measured, |AE |2�E AE |2 gives the probability of getting the value E, if ψ is
not normalized. (There is no need to divide by the denominator if ψ is normalized.)
With a lot more machinery we can reduce the number of postulates by one, but it is not
worth it. This was discussed in the optional Section VI.
Postulate 7 The evolution of ψ with time is given by the Schrodinger equation
i�∂ψ(x, t)
∂t=
�− h
2
2m
∂ψ(x, t)
∂x2+ V (x)ψ(x, t)
�(159)
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where V (x) is the potential at point x.
A corollary is that if at t = 0
ψ(x, 0) =�
E
AE(0)ψE(x),
then at any future time t,
ψ(x, t) =�
E
AE(0)e−iEt/�ψE(x)
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