Quantum Mechanics - Prof. Eric R. Bittner - University of Houston

Z'
Y
Z
Y'
where we notice that −p 2 φ/mr 3 is the centrifugal force. Taking the last equation, multiplying by
˙r and then integrating with respect to time gives
i.e.
X'
˙r 2 = − p2 φ
m 2 r 2 − kr2 + b (1.21)
˙r =
Integrating once again with respect to time,
�
t − to =
=
X
− p2 φ
m 2 r 2 − kr2 + b (1.22)
� rdr
�
= 1
�
2
˙r
�
rdr
− p2 φ
m 2 − kr 4 + br 2
dx
√ a + bx + cx 2
(1.23)
(1.24)
(1.25)
where x = r 2 , a = −p 2 φ/m 2 , b is the constant of integration, and c = −k This is a standard
integral and we can evaluate it to find
where
r 2 = 1
2ω (b + A sin(ω(t − to))) (1.26)
�
A =
m2 .
What we see then is that r follows an elliptical path in a plane determined by the initial velocity.
b 2 − ω2 p 2 φ
22