Chapter 7 Lecture 7 - Energy Eigenstates and Eigenvalues

7.1 Time Dependence

We saw that (from equations II.25 and II.26)

¶ t

p_x=i

¶ x

are plasusible operators for energy and momentum. Assuming these are generally true we can write down an energy eigenvalue equation.

EY (x,y,z,t)=EY (x,y,z,t)

¶ t

Y (x,y,z,t)=EY (x,y,z,t) (7.1)

From equation II.27 we know that E~¶/¶ t acts on the time dependent part of the wavefunction. The spacial part of the energy eigenfunction will be independent of t and therefore we can seperate out time dependent parts of Y (x,y,z,t)

Y (x,y,z,t)=U(x,y,z)T(t) (7.2)

where U(x,y,z) is spatial whilst T(t) is temporal. eg. for plane waves

(kx-w t)

-w t

from equation II.28 we have

¶ t

(U(x,y,z)T(t))=U(x,y,z)

¶ t

T(t)=EU(x,y,z)T(t)

dividing both sides by U(x,y,z)

¶ t

T(t) (7.3)

the general solutions are of the form

T(t)=Ae

(7.4)

The energy of an energy eignestate determines the temporal evolution.

7.2 An Energy Operator in Spatial Coordinates

In classical physics

E=E_{kinetic energy}+V(x,y,z) (7.5)

where V(x,y,z) is a potential function

E_{kinetic energy}=

mv²=

p²

(7.6)

we invoke the Correspondence Principle. ie. equivalence of classical and quantum physics under appropiate limits. so converting from classical to quantum physical terms

p®p

E_{kinetic energy}®

p²

V(x,y,z)®V(x,y,z) (7.7)

so p in three dimensions

¶ x

® Ñ

where

Ñ=i

¶ x

¶ y

¶ z

therefore

p=Ñ (7.8)

(p)²

(-Ñ)²

Ñ² (7.9)

where

Ñ²=

¶²

¶ x

¶²

¶ y

¶²

¶ z

(in Cartesian form) (7.10)

Thus we have an operator for energy called the Hamiltonian Operator

H=-

Ñ²+V(x,y,z) (7.11)

HY (x,y,z,t)=EY (x,y,z,t)=

¶ t

Y (x,y,z,t)

¶ t

Y (x,y,z,t)=

é
ê
ê
ë

Ñ²+V

ù
ú
ú
û

Y (x,y,z,t) (7.12)

Equation II.38 is the Time-Dependent Schrödinger equation. The solutions of this equation are wavefunctions of the states of definite energies (the energy eigenstates). The outcomes of this equation are

this energy eigenvalue equation is closely related to a wave equation
it gives the probability amplitude for position measurements (wave function) Y (r,t) with probability density Y^*(r,t)Y (r,t) .

H acts only on the spatial part of the wavefunction

Y (x,y,z,t)=U(x,y,z)T(t)=U(x,y,z)e

Y (x,y,z,t)=e

U(x,y,z)=e

EU(x,y,z) (7.13)

Thus for an energy eigenvalue E_n we have for the eigenstate U_n the following Time Independent Schrödinger Equation

é
ê
ê
ë

Ñ²+V(x,y,z)

ù
ú
ú
û

U_n(x,y,z)=E_nU_n(x,y,z) (7.14)