Download Quantum Mechanics

量子力學導論
Chap
Chap
Chap
Chap
1
2
3
4
- The Wave Function
- The Time-independent Schrödinger Equation
- Formalism in Hilbert Space
-表象理論
Quantum Mechanics
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
finding the particle
inside the well V = 0
thus
, probability is zero for
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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 < 0
similarly
use boundary condition :
find k : from
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and
so
normalize
and
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(ii) scattering state : E > 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