Linearity of the Schrödinger Equation Linearity of the Schrödinger

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.
E3
!1(x,t) is a solution and thus satisfies:
E1
!2(x,t) is a solution and thus satisfies:
Add Eqs. E1 and E2 together as c1E1+c2E2:
Rearrange a bit:
Differentiation is linear:
Substitute Eqn. E3 to recover the Schrödinger equation
for !(x,t) thus showing that !(x,t) is also a solution.
Linearity of the Schrödinger Equation
Example: Electron Double Slit Experiment:
(1) Two electron waves:
z
(2) Superposition of waves:
x
Caution: The above is a simplified plausibility argument,
proper treatment requires wavepackets and consideration
of "kx !!
E2
The Time-Dependent Schrödinger Equation
An operator equation acting on !(x,t)
Classical
Rearrange equation:
Quantum Mechanics
p2
p
Drop ! on both sides to obtain an operator equation:
x
Compare terms with classical energy expression:
E
H
Separation of Variables
A 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):
Partial differential equation
Use:
Separable, because equation
has to hold for all x and all y.
Ordinary differential
equation for h(y) and
its solution.
Ordinary differential
equation for g(x) and
its solution.
Combine solutions:
A,B,C,D are
arbitrary
constants.
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 timeindependent or stationary.
Required Properties of Eigenfunctions
#(x) and d#(x)/dx must be finite, continuous and single valued.
Examples of invalid forms of #(x) and d#(x)/dx:
This creates constraints on physically allowable solutions, which in turn produces
quantization for certain types of potentials V(x).
Qualitative Link between V(x) and #(x)
E
#(x)
#(x)
d2#/dx2>0
d2#/dx2<0
x