苏汝铿高等量子力学讲义(英文版)Chapter 1 Foundation of Quantum Mechanics

§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
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)
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
§1.3 Operators
Commutator
§1.3 Operators
Commutator
§1.3 Operators
Commutator
百度文库
§1.3 Operators
Hermitian operator
§1.3 Operators
Eigenequation
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)
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
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 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)
Conservation of the probability
§1.5 WKB method (Wentzel-Kramers-Brillouin)
E = U(x) Turning points: The semiclassical approximation is not applicable
§1.5 WKB method (Wentzel-Kramers-Brillouin)
§1.3 Operators
O - representation
§1.3 Operators
O - representation
§1.4 Approximation method
Perturbation independent of time Non -degenerate
§1.4 Approximation method
§1.5 WKB method (Wentzel-Kramers-Brillouin)
x1 B exp{− ∫ κ ( y )dy} at a1 x 2 p ψ ( x) = B sin{ x k ( y )dy + π } at b 1 ∫x1 p 4
§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
§1.2 Schrödinger equation and its solutions
Coulomb potential
§1.2 Schrödinger equation and its solutions
Coulomb potential
§1.3 Operators
According to the Born statistical interpretation, The probability of finding a particle at position r is just the square of its wave function
[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)
§1.2 Schrödinger equation and its solutions
Infinite potential well
§1.2 Schrödinger equation and its solutions
Harmonic oscillator
§1.2 Schrödinger equation and its solutions
§1.4 Approximation method
Perturbation depending on time
§1.4 Approximation method
Perturbation depending on time
§1.4 Approximation method
Variational method Key: How to choose the trial wave function
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)
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)
§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
§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
§1.6 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)
Harmonic oscillator
§1.2 Schrödinger equation and its solutions
3D Schrodinger equation Central potential
§1.2 Schrödinger equation and its solutions
Central potential
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) ∝ p
2
§1.5 WKB method (Wentzel-Kramers-Brillouin)
§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
§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)
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)
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