MORE CHAPTER 6, #3
Schrödinger’s Trick
The time-dependent Schrödinger equation for the harmonic oscillator is
-
U2 0 2
1
0
+ Kx 2 = iU
2m 0 x 2
2
0t
[1]
whose stationary, bound-state solutions are
1x,t2 = 1x2e -iEt>U
where (x) satisfies the time-independent equation
-
U2 d 21x2
1
+ Kx 21x2 = E1x2
2
2m dx
2
[2]
It is not obvious how to solve Equation 2 for the allowed values of E and the
corresponding wave functions (x). There are several general techniques for solving
differential equations; however, this problem can be solved (exactly!) using a beautiful trick invented by Schrödinger.
Recalling that = 2K>m, we define y = 2m>Ux and, correspondingly,
dy = 2m>U dx. Note that is the classical oscillator’s angular frequency:
x = x0 cos t, which satisfies m 1d 2 x>dt 2 2 = -Kx. Therefore, substituting x and dx
in terms of y and dy from above into Equation 2, we obtain
and
d 2
U2
1
1
U 2 2
2
+
1m
2
a
b y = E
2m 1 2U>m 2 2 dy 2
2
A m
d 2
2E
- y 2 = or
2
U
dy
This can be written as
c
2E
d2
- y2 d = 2
U
dy
d
2E
d
- yb a
+ yb - 1 d = dy
dy
U
To see that this is true, note that
ca
a
d
d
d
d
- yb a
+ yb - = a
- yb a
+ yb - dy
dy
dy
dy
=
26
d
d
d 2
d 2
+ y
+ y - y 2 - =
- y
- y 2
2
dy
dy
dy
dy 2
[3]
[4]
More Chapter 6
So the Schrödinger equation for the harmonic oscillator becomes
a
d
d
2E
- yb a
+ yb = a 1 b
dy
dy
U
d
Operating on Equation 5 from the left with a
+ yb , we obtain
dy
d
d
d
2E
d
a
+ yb a
- yb a
+ yb = a 1 ba
+ yb dy
dy
dy
U dy
[5]
But, for any function f
a
df
d
d
d
- yb a
+ yb f = a
- yb a
+ yfb
dy
dy
dy
dy
d2f
df
df
d2
2
+
y
y
f
y
f
=
a
- y 2 - 1b f
dy
dy
dy 2
dy 2
d
This is true for any function f(y), in particular for f1y2 = a
+ yb . Therefore,
dy
=
a
d2
d
d
2E
d
- y2b a
+ yb - a
+ yb = a 1 ba
+ yb 2
dy
dy
U dy
dy
Rearranging this gives us
a
21E - U2
d2
d
d
- y2b c a
+ yb d = ca
+ yb d
dy
U
dy
dy 2
[6]
But recalling Equation 3, which is
c
d2
2E
- y2 d = U
dy 2
d
we see that, if we define = a
+ yb and E = E - U, then Equation 6
dy
becomes Equation 7:
c
d2
2E
- y 2 d = 2
U
dy
[7]
Thus, Equations 3 and 7 have the exact same form. This means that if we have found
a solution (y) corresponding to energy E, then ((d>dy) + y) = (d>dy) + y is
also a solution, and its corresponding energy will be 1E - U2. We can just keep
going like this and each time the energy is lowered by U. This means that the spacing of the energy levels of the quantum harmonic oscillator is U.
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