bei48482_FM

Before we tackle Schrödinger’s equation, let us review the wave equation

Wave equation  (5.3)

which governs a wave whose variable quantity is ythat propagates in the xdirection

with the speed . In the case of a wave in a stretched string, yis the displacement of

the string from the xaxis; in the case of a sound wave, yis the pressure difference; in

the case of a light wave, yis either the electric or the magnetic field magnitude.

Equation (5.3) can be derived from the second law of motion for mechanical waves

and from Maxwell’s equations for electromagnetic waves.

^2 y

t^2

1

^2

^2 y

x^2

164 Chapter Five

Partial Derivatives

S

uppose we have a function f(x, y) of two variables, xand y, and we want to know how f

varies with only one of them, say x. To find out, we differentiate fwith respect to xwhile

treating the other variable yas a constant. The result is the partial derivativeof fwith respect

to x, which is written fx

  

yconstant

The rules for ordinary differentiation hold for partial differentiation as well. For instance, if

fcx^2 ,

 2 cx

and so, if fyx^2 ,

  

yconstant

 2 yx

The partial derivative of fyx^2 with respect to the other variable, y, is

  

xconstant

x^2

Second order partial derivatives occur often in physics, as in the wave equation. To find

^2 fx^2 , we first calculate fxand then differentiate again, still keeping yconstant:

 

For fyx^2 ,

 (2yx) 2 y

Similarly  (x^2 ) 0



(^) y
^2 f
(^) y 2

x
^2 f
x^2
f
x

x
^2 f
x^2
df
dy
f
y
df
dx
f
x
df
dx
df
dx
f
x
Solutions of the wave equation may be of many kinds, reflecting the variety of
waves that can occur—a single traveling pulse, a train of waves of constant amplitude
and wavelength, a train of superposed waves of the same amplitudes and
wavelengths, a train of superposed waves of different amplitudes and wavelengths,
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