We therefore write (3.12) as
h
2
ih ¯
∂ ψ(x, t) = −
¯ v
2 ψ(x, t) + q φ(x) ψ(x, t).
(3.13)
∂t
2m
This is known as the time-dependent Schr¨ odinger equation. It is a
linear partial differential equation of second order in x, and first
order in t. It applies to general curvilinear coordinates as well as
Cartesian coordinates, where one substitutes the appropriate form
for the Laplacian operator v
2 . It can be solved in principle for the
eigenfunction ψ(x, t), given the explicit form for φ and appropriate
boundary conditions.
We now investigate the physical meaning of the eigenfunction
ψ(x, t). We assume that ψ can be written in the separable form
ψ(x, t) = u(x) τ (t),
(3.14)
where u is a function only of x and τ is a function only of t.
The function u is not to be confused with the complex transverse
particle position in the earlier description of classical geometrical
optics. Substituting above, and dividing through by uτ , we obtain
h
2
1 d
1
¯
ih ¯
τ (t) =
−
v
2 u(x) + q φ(x) = H. (3.15)
τ (t) dt
u(x)
2m
We notice that the left side depends only on t, while the middle
depends only on x. This is only true in the case where φ is independent of time, which we assume for now to be the case. This
separation of variables can only hold for all x and t if both sides
are equal to an arbitrary constant, which we call H. The physical meaning of H will become apparent in the following, but for
now it is just an arbitrary constant. Substituting, we obtain two
separate, decoupled equations given by
d
iH
τ (t) +
τ (t) = 0
dt
h ¯
2m
v
2 u(x) +
[ H − qφ(x) ] u(x) = 0.
(3.16)
h
2
¯
3.1. Quantum mechanical description of particle motion
129