(a) Show, for an appropriate value of a which you should find, that Vo(x) = Ae-α2² is an eigenfunction of the time-independent Schrödinger equation with potential V(x) = 1/mw²x². Hence find the associated energy eigenvalue. Using the standard integral find the value of A. L - Mx² dx = M M > 0,
(a) Show, for an appropriate value of a which you should find, that Vo(x) = Ae-α2² is an eigenfunction of the time-independent Schrödinger equation with potential V(x) = 1/mw²x². Hence find the associated energy eigenvalue. Using the standard integral find the value of A. L - Mx² dx = M M > 0,
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Transcribed Image Text:(a) Show, for an appropriate value of a which you should find, that
Vo(x) = Ae-αx²
is an eigenfunction of the time-independent Schrödinger equation with
potential
1
V(x) = -mw²x².
Hence find the associated energy eigenvalue.
Using the standard integral
L
-Mr² dx
ㅠ
M'
M > 0,
find the value of A.
(b) The ladder operators associated with a particle under the potential V(x),
defined in part (a), are given by
1
â=
=
= √ (2 + 1²² ) , &t = √ (2-14).
dx
dx
where L²= h/mw.
i. Show that âo(x)=0 where Vo(x) is the function in part (a). What
is the physical interpretation of this result?
ii. Derive the commutation relations for the ladder operators â and at.
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