[b2,a2] region
x D exp{− ∫ κ ( x ')dx '} at a2 x2 2 p ψ ( x) = D sin{ x2 k ( x ')dx ' + π } at b 2 ∫x p 4
§1.5 WKB method (Wentzel-Kramers-Brillouin)
Quantum Mechenics II
Ru-Keng Su
2005.1.5
Chapter 1 Foundation of Quantum Mechanics
§1.1 State vector, wave function and superposition of states
This chapter evolves from an attempt of a brief review over the basic ideas and formulae in undergraduate-level quantum mechanics. The details of this chapter can be found in the usual references of quantum mechanics
§1.3 Operators
O - representation
§1.3 Operators
O - representation
§1.4 Approximation method
Perturbation independent of time Non -degenerate
§1.4 Approximation method
E = U(x)
§1.5 WKB method (Wentzel-Kramers-Brillouin)
E = U(x)
§1.5 WKB method (Wentzel-Kramers-Brillouin)
E < U(x)
§1.5 WKB method (Wentzel-Kramers-Brillouin)
Non -degenerate
§1.4 Approximation method
Non -degenerate
§1.4 Approximation method
Degenerate
§1.4 Approximation method
Degenerate
§1.4 Approximation method
§1.5 WKB method (Wentzel-Kramers-Brillouin)
§1.5 WKB method (Wentzel-Kramers-Brillouin)
§1.5 WKB method (Wentzel-Kramers-Brillouin)
For 1D case
§1.5 WKB method (Wentzel-Kramers-Brillouin)
Harmonic oscillator
§1.2 Schrödinger equation and its solutions
Harmonic oscillator
§1.2 Schrödinger equation and its solutions
Harmonic oscillator
§1.2 Schrödinger equation and its solutions
Example I:
§1.5 WKB method (Wentzel-Kramers-Brillouin)
E < U(x)
§1.5 WKB method (Wentzel-Kramers-Brillouin)
E > U(x)
§1.5 WKB method (Wentzel-Kramers-Brillouin)
§1.4 Approximation method
Variational method
§1.5 WKB method (Wentzel-Kramers-Brillouin)
Basic idea: (Q.M.) (C.M) when h 0 WKB Semi- Classical method: To find an expansion of h and solve stationary Schrödinger equation
§1.3 Operators
§1.3 Operators
Commutator
§1.3 Operators
Commutator
§1.3 Operators
Commutator
§1.3 Operators
Hermitian operator
§1.3 Operators
Eigenequation
1 b= 2m[U (l ) − E ] h
1 l γ = ∫ 2m[U ( x) − E ]dx h 0
§1.5 WKB method (Wentzel-Kramers-Brillouin)
Barrier penetration
§1.5 WKB method (Wentzel-Kramers-Brillouin)
§1.6 Density matrix
Problem: Can we get a new formula to calculate the expectation value like quantum statistics Q.M. <A> = <n|A|n> Q.S. <A> = tr (ρA) = tr (exp(-βH)A)
§1.2 Schrödinger equation and its solutions
§1.2 Schrödinger equation and its solutions
§1.2 Schrödinger equation and its solutions
1D Schrödinger equation Infinite potential well
E > U(x)
§1.5 WKB method (Wentzel-Kramers-Brillouin)
[a1,b1] region
§1.5 WKB method (Wentzel-Kramers-Brillouin)
E > U(x)
Asymptotic solutions
§1.5 WKB method (Wentzel-Kramers-Brillouin)
Advantages of this choice are
§1.4 Approximation method
Degeneracy may be removed
§1.4 Approximation method
Perturbation depending on time Key: How to calculate the transition amplitude
§1.3 Operators
§1.3 Operators
§1.3 Operators
§1.3 Operators
pi -ih/2π▽i Cartesian rectangular coordinates 1st convention: pure coordinate part pure momentum part 2nd convention: mixed part
Barrier penetration
§1.5 WKB method (Wentzel-Kramers-Brillouin)
Connection formulae (dU/dx>0)
§1.5 WKB method (Wentzel-Kramers-Brillouin)
Connection formulae (dU/dx<0)
For 1D case
§1.5 WKB method (Wentzel-Kramers-Brillouin)
For 1D case
§1.5 WKB method (Wentzel-Kramers-Brillouin)
Three regions: E > U(x)
1 ψ ( x) ∝ p2来自§1.5 WKB method (Wentzel-Kramers-Brillouin)
Barrier penetration
§1.5 WKB method (Wentzel-Kramers-Brillouin)
Barrier penetration
α eγ + β e −γ = C beik0t ik t γ −γ b [α e − β e ] = ik0Ce 0
§1.1 State vector, wave function and superposition of states
§1.1 State vector, wave function and superposition of states
§1.1 State vector, wave function and superposition of states
This is the Bohr-Sommerfeld quantized condition
§1.5 WKB method (Wentzel-Kramers-Brillouin)
Example 2: Barrier penetration
§1.5 WKB method (Wentzel-Kramers-Brillouin)
§1.2 Schrödinger equation and its solutions
Infinite potential well
§1.2 Schrödinger equation and its solutions