Chap 1 The Wave Function

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Chap 1 - The Wave Function Chap 2 - The
Time-independent Schrödinger Equation Chap 3 -
Formalism in Hilbert Space Chap 4 -????
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Chap 2 - The Time-independent Schrödinger
Equation
? 2.1 Stationary state
Assume V is independent of t , use separation of
variables
Deduce from equation (2.1) , then
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...(2.2)
time-independent Schrödinger equation
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properties of
(i) Stationary state
for every expectation value
is constant in time
(ii) Definite total energy
Classical mechanics total energy is Hamiltonian
Quantum mechanics corresponding Hamiltonian
operator
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thus equation (2.2)
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(iii) Linear combination of separable solution
? 2.2 Infinite square well( one dimensional box)

and boundary conditions
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deduce
outside the potential well ,
probability is zero for finding the particle
inside the well V 0
thus
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normalize

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first three states and probability density of
infinite square well

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? 2.3 Harmonic oscillator
Classical treatment
solution
potential energy V is related to F
Quantum treatment
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solve equation (2.3) by use ladder operator
rewrite equation (2.3) by ladder operator
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compare equation(2.3)
similarly
discussion (i)
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and

(ii)
and
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and

(iii) there must exist a min state with
and from
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and

so the ladder of stationary states can
illustrate
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and

? 2.4 Delta-function potential
Energy E
Consider potential
then
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(i) bound state E lt 0

similarly
use boundary condition
find k from
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and
so
and
normalize
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(ii) scattering state E gt 0
and
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for wave coming in from left ( D 0 ) , equation
(2.5),(2.6) rewrite
is incident wave
is reflected wave
is transmitted wave
R is reflection coefficient
T is transmission coefficient
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? 2.5 Free particle

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Fourier transform
wave packet moves along at group velocity
and phase velocity
so
wave packet