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34
CHAPTER 3. BASIC IDEAS OF QUANTUM MECHANICS
The other eigenfunctions do not necessarily have their maximum magnitude at the origin: for
example, the shown states
ψ
100
and
ψ
010
are zero at the origin.
For large negative values of its argument, an exponential becomes very small very quickly. So
if the distance from the origin is large compared to
ℓ
, the wave function will be negligible, and
it will be zero in the limit of in±nite distance.
For example, if the distance from the origin is just 10 times
ℓ
, the exponential above is already
as small as 0.000,000,000,002 which is clearly negligible.
As far as the value of the other eigenfunctions at large distance from the origin is concerned,
note from table 3.1 that all eigenfunctions take the generic form
ψ
n
x
n
y
n
z
=
polynomial in
x
e
x
2
/
2
ℓ
2
polynomial in
y
e
y
2
/
2
ℓ
2
polynomial in
z
e
z
2
/
2
ℓ
2
.
For the distance from the origin to become large, at least one of
x
,
y
, or
z
must become large,
and then the blow up of the corresponding exponential in the bottom makes the eigenfunctions
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 Fall '11
 Dr.DanielArenas
 mechanics

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