E1E2

Linearity of the Schrödinger Equation

Linearity in !(x,t): A linear combination !(x,t) of two solutions !1(x,t) and !2(x,t) is

also a solution.

Rearrange a bit:

Add Eqs. E1 and E2 together as c1E1+c2E2:

!1(x,t) is a solution and thus satisfies: !2(x,t) is a solution and thus satisfies:

Differentiation is linear:

E3

Substitute Eqn. E3 to recover the Schrödinger equationfor !(x,t) thus showing that !(x,t) is also a solution.

Linearity of the Schrödinger Equation

Example: Electron Double Slit Experiment:

x

z

Caution: The above is a simplified plausibility argument,

proper treatment requires wavepackets and consideration

of "kx !!

(1) Two electron waves:

(2) Superposition of waves:

The Time-Dependent Schrödinger Equation

An operator equation acting on !(x,t)

Compare terms with classical energy expression:

Drop ! on both sides to obtain an operator equation:

Rearrange equation:

p

H

E

x

p2

Quantum MechanicsClassical

Separation of VariablesA mathematical trick to split a partial differential equation (in several variables) into

several ordinary differential equations (in a single variable each).

Simple abstract example (of no physical relevance):

Combine solutions:

Ordinary differential

equation for g(x) and

its solution.

Ordinary differential

equation for h(y) and

its solution.

Separable, because equation

has to hold for all x and all y.

Use:

A,B,C,D are

arbitrary

constants.

Partial differential equation

The Time-Independent Schrödinger Equation

For a time-independent potential: Search for product solutions:

Inserted into the time-dependent Schrödinger equation and separation of variables

gives two ordinary differential equations in x and t:

(the time-independent Schrödinger equation)

General form of the wavefunction

For a time-independent potential V(x).

For time-independent potential, the probability function P=|!(x,t)|2 is time-

independent or stationary.

Required Properties of Eigenfunctions

#(x) and d#(x)/dx must be finite, continuous and single valued.

This creates constraints on physically allowable solutions, which in turn produces

quantization for certain types of potentials V(x).

Examples of invalid forms of #(x) and d#(x)/dx:

Qualitative Link between V(x) and #(x)

E

#(x)

x

#(x)d2#/dx2>0

d2#/dx2<0