Simple Nature - Light and Matter

physics the wavefunction will always have a tail that reaches into

the classically forbidden region. If it was not for this effect, called

tunneling, the fusion reactions that power the sun would not occur

due to the high electrical energy nuclei need in order to get close

together! Tunneling is discussed in more detail in the following

subsection.

13.3.6 The Schrodinger equation ̈

In subsection 13.3.5 we were able to apply conservation of energy

to an electron’s wavefunction, but only by using the clumsy graph-

ical technique of osculating sine waves as a measure of the wave’s

curvature. You have learned a more convenient measure of curvature

in calculus: the second derivative. To relate the two approaches, we

take the second derivative of a sine wave:

d^2

dx^2

sin(2πx/λ) =

d

dx

(

2 π

λ

cos

2 πx

λ

)

=−

(

2 π

λ

) 2

sin

2 πx

λ

Taking the second derivative gives us back the same function,

but with a minus sign and a constant out in front that is related

to the wavelength. We can thus relate the second derivative to the

osculating wavelength:

[1]

d^2 Ψ

dx^2

=−

(

2 π

λ

) 2

Ψ

This could be solved forλin terms of Ψ, but it will turn out

below to be more convenient to leave it in this form.

Applying this to conservation of energy, we have

E=K+U

=

p^2

2 m

+U

=

(h/λ)^2

2 m

+U

[2]

Note that both equation [1] and equation [2] haveλ^2 in the denom-

inator. We can simplify our algebra by multiplying both sides of

equation [2] by Ψ to make it look more like equation [1]:

904 Chapter 13 Quantum Physics