Simple Nature - Light and Matter

E·Ψ =

(h/λ)^2

2 m

Ψ +U·Ψ

=

1

2 m

(

h

2 π

) 2 (

2 π

λ

) 2

Ψ +U·Ψ

=−

1

2 m

(

h

2 π

) 2

d^2 Ψ

dx^2

+U·Ψ

Further simplification is achieved by using the symbol~(hwith a

slash through it, read “h-bar”) as an abbreviation forh/ 2 π. We then

have the important result known as theSchr ̈odinger equation:

E·Ψ =−

~^2

2 m

d^2 Ψ

dx^2

+U·Ψ

(Actually this is a simplified version of the Schr ̈odinger equation,

applying only to standing waves in one dimension.) Physically it

is a statement of conservation of energy. The total energyEmust

be constant, so the equation tells us that a change in interaction

energyU must be accompanied by a change in the curvature of

the wavefunction. This change in curvature relates to a change

in wavelength, which corresponds to a change in momentum and

kinetic energy.

self-check G

Considering the assumptions that were made in deriving the Schrodinger ̈

equation, would it be correct to apply it to a photon? To an electron mov-

ing at relativistic speeds? .Answer, p.

1063

Usually we know right off the bat howU depends onx, so the

basic mathematical problem of quantum physics is to find a function

Ψ(x) that satisfies the Schr ̈odinger equation for a given interaction-

energy functionU(x). An equation, such as the Schr ̈odinger equa-

tion, that specifies a relationship between a function and its deriva-

tives is known as a differential equation.

The detailed study of the solution of the Schr ̈odinger equation

is beyond the scope of this book, but we can gain some important

insights by considering the easiest version of the Schr ̈odinger equa-

tion, in which the interaction energyU is constant. We can then

rearrange the Schr ̈odinger equation as follows:

d^2 Ψ

dx^2

=

2 m(U−E)

~^2

Ψ,

which boils down to

d^2 Ψ

dx^2

=aΨ,

Section 13.3 Matter as a wave 905