130_notes.dvi

1.10 A Particle in a Box

As a concrete illustration of these ideas, we study the particle in a box (See section 8.5) (in one
dimension). This is just a particle (of massm) which is free to move inside the walls of a box
0 < x < a, but which cannot penetrate the walls. We represent that by a potential which is zero
inside the box and infinite outside. We solve theSchr ̈odinger equationinside the box and realize
that the probability for the particle to be outside the box, and hence the wavefunction there, must
be zero. Since there is no potential inside, the Schr ̈odinger equation is

Hun(x) =−

̄h^2

2 m

d^2 un(x)

dx^2

=Enun(x)

where we have anticipated that there will be many solutions indexed byn. We know four (only 2
linearly independent) functions which have a second derivative whichis a constant times the same
function:u(x) =eikx,u(x) =e−ikx,u(x) = sin(kx), andu(x) = cos(kx). The wave function must
be continuous though, so we require the boundary conditions

u(0) =u(a) = 0.

The sine function is always zero atx= 0 and none of the others are. To make the sine function zero
atx=awe needka=nπork=nπa. So theenergy eigenfunctionsare given by

un(x) =Csin

(nπx

a

)

where we allow the overall constantCbecause it satisfies the differential equation. Plugging sin

(nπx

a

)

back into the Schr ̈odinger equation, we find that

En=

n^2 π^2 ̄h^2

2 ma^2

Onlyquantized energiesare allowed when we solve this bound state problem. We have one
remaining task. The eigenstates should be normalized to representone particle.

〈un|un〉=

∫a

0

C∗sin

(nπx

a

)

Csin

(nπx

a

)

dx=|C|^2

a

2

So the wave function will be normalized if we chooseC=

√

2

a.

un(x) =

√

2

a

sin

(nπx

a

)

We can always multiply by any complex number of magnitude 1, but, it doesn’t change the physics.
This example shows many of the features we will see for other boundstate problems. The one
difference is that, because of an infinite change in the potential at the walls of the box, we did not
need to keep the first derivative of the wavefunction continuous.In all other problems, we will have
to pay more attention to this.

1.11 Piecewise Constant Potentials in One Dimension

We now study the physics of severalsimple potentials in one dimension. First a series of
piecewise constant potentials (See section 9.1.1). for which the Schr ̈odinger equation is

− ̄h^2

2 m

d^2 u(x)

dx^2

+V u(x) =Eu(x)