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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