Chem 120A
Problem Set 2
Out: February 14, 2006; Due: February 22, 2006
1. a) If
A
is an operator with
A
≡
±
±
α
² ³
β
±
±
, prove that
A
†
=
±
±
β
² ³
α
±
±
.
b) Show that the product of two Hermitian operators need not be Hermitian.
Under what
condition is the product of two Hermitian operators Hermitian?
2. The energy di±erence between the ground state (lowest energy level) and ²rst excited state
(next level up) in a hydrogen atom is about 10 eV (eV = electron volt). The diameter of a
hydrogen atom is
≈
1
˚
A= 10

10
m.
If we model a hydrogen atom as a 1D box with hard
walls, then what is the length of the box to get the same energy level spacing as hydrogen?
3. Consider a particle of mass = m sitting in the ground state of a box of length=l.
Suppose
that one wall of the box is
suddenly
moved out so that the length of the box becomes length
= 3l.
a) If the energy of the particle is measured right after moving the wall, then what is the
probability that the particle with be found in the n=10 state of the new box?
b) How does this probability change with time?
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 Spring '07
 Whaley
 Atom, Neutron, Hydrogen atom, 2m, CHEM 120A

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