but ò -¥¥ [-i hbar(d/dx)y*y dx = 0 [see below], so we must have
<p>= ò[y*(-i hbar)(d/dx)y dx]
To maintain consistency with classical physics, we must preserve the relation E = px2 / (2m) + V(x). The easiest way to preserve this relation is to relate the momentum and position operators in the same way. Hence we want
[pop2 / (2m)]y
+ V(x)y
= Eopy
Substituting the relations for operators acting on y gives the classical relation multiplied by y, which is good.
Substituting for the operators
[- hbar2/(2m)] (d2/dx2)y + V(x)y = i hbar (d/dt)y [2-1]
This equation should be regarded as a guess, to be confirmed or contradicted by the success of its predictions. We will have to calculate some solutions to see if the equation is correct.
Equation 2-1 is the Schroedinger Equation. It is the F=ma of Nonrelativistic Quantum Mechanics. It is frequently written more compactly by taking
H = - [hbar2/(2m)](d2/dx2) + V(x) [2-1']
Hy = i hbar dy/dt
We must have
1 = ò -¥¥ [y*(x,t)y(x,t) dx] [3-1]
regardless of t, so that there will be a probability of one that there will be a particle somewhere at any time. Since y is a solution of a homogeneous differential equation, it will always have an overall arbitrary constant. This constant can be used at any one arbitrary time to satisfy eq. 3-1, provided that the integral in that equation is finite. Hence to have acceptable physics, we must have that y(x,t) goes to zero faster than x -1/2 at large |x| for any given t.
If eq. 3-1 is satisfied at any given time, and if differentiating eq. 3-1 with respect to time gives zero, the normalization condition will hold forever. So we need
0 = (d/dt) ò -¥¥ [y*(x,t)y(x,t) dx].
But is it? From p. 12, we have
(d/dt) ò -¥¥
[y*(x,t)y(x,t) dx]
= ò -¥¥
[(dy/dt)*y
+y*(dy/dt)] dx
Since (d/dt)y
= [i hbar/(2m)]d2y/dx2
-(i/ hbar) Vy, implying also that
(d/dt)y*
= [-i hbar/(2m)]d2y*/dx2
+(i/ hbar) Vy*
[V is real], we have
(d/dt) ò -¥¥
[y*(x,t)y(x,t) dx]
= [i hbar/(2m)]
ò -¥¥
[y* d2y/dx2
- (d2y*/dx2)y dx]
= [i hbar/(2m)]
ò -¥¥
(d/dx) [y* dy/dx
- (dy*/dx)y dx]
= [i hbar/(2m)]
[y* dy/dx
- (dy*/dx)y
] -¥¥
Now we have already had to require that y goes to zero faster than x -1/2 for large x. Therefore [...] -¥ ¥=0, and
(d/dt) ò -¥¥
[y*(x,t)y(x,t) dx]=0
and the total probability of finding a particle is 1 at all times, as
required for consistent physics.
In summary, for one dimension the boundary condition y(x, t) falls faster than x -1/2 for large |x| and all t is necessary and sufficient to guarantee that the wave function may be properly normalized. [In three dimensions the situation is slightly more delicate. We will face this problem only when necessary.]
| Last revised 2005/02/11 |