∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧
∧
∧
∧
∧
4.1-3 Bra and Ket Notation(左矢和右矢符号)
A scalar product of the square-integrable function can be expressed
∧
∧
∧
∧
∧
∧
∧
∧
∧
∧
∧
∧
- zy p x p z + z p y p y + yx p z p z + x( p z z ) p y xz p z p y
2
∧
∧
∧
∧
∧
∧
∧
∧
∧
∧
= ihy p x + ihx p y = ih ( x p y y p x ) = ih L z
∧ ∧ ∧ ∧ ∧
∧
∧
+∞
∞
ψ 1ψ 2 dV = ψ 1 ψ 2 ∫
ψ 1 is called a " bra" (左矢), 2 is called a " ket" (右矢) ψ
ψ
= ψ
The orthonormality relation of two wave functions
+∞ ψ mψ n dV= ψ m ψ n =δ mn ∫
Chapter 4 Mathematics Foundations of Quantum Mechanics I
(量子力学中的数学基础)
4.1 Properties of Operaotors
What is operator? Operator in quantum mechanics denotes an operation of wave function, such as
x p x ψ = ihx ψ x
∧
p x xψ = ih ( xψ ) = ihψ ihx ψ x x
∧
So
( x p x p x x ) = ih ≠ 0
∧
∧
We can similarly obtain
( y p y p y y ) = ih
But
∧
∧
( z p z p z z ) = ih ( x p z p z x) = 0
Linear operator is self-adjoint (自共轭) or Hermitian (厄密的)
ψ Lψ 2dV = ∫ ( Lψ 1 )ψ 2dV ∫
1
∧
∧
In quantum mechanics, all operates are Hermitian operators
L = ∫ψ Lψ dV = ∫ (ψ L) ψ dV = [ ∫ψ ( Lψ )dV ] = L
∧ ∧ ∧
[ L, H ] = 0
[H , L ] = 0
∧
∧2
4.2 Eigenvalue and Eigenfunction (本征值和本征函数 本征值和本征函数)
If L is a constant value, its deviation L=0, so we can find the corresponding wave function ψL
∧ ∧
∧
∧
( x p y p y x) = 0
In summary
∧ ∧
∧
∧
( xα p β p β xα ) = ihδ αβ , α , β = x,y,z
if AB - BA = 0, A and B are commuting, we mark it
AB - BA = [A, B] , or [A, B]
∞
The mean value of L
L = ψ L ψ = ∫ψ ψ dV
∧
4.1- 4 Operator in quantum mechanics
(1) position and momentum operators (P73) Position operator
r=r
Its components
∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧2
∧2 ∧
∧
∧2
∧
∧
∧
∧
∧2
∧
∧
∧
∧
∧
∧2
The second term
( L z L x - L x L z ) = L z ( L z L x ) - L x L z = L z (ih L y + L x L z ) - L x L z = ih L z L y + ( L z L x ) L z - L x L z
∧
∧
(2) Angular-momentum operators (角动量算符) P75
L = r× p = ih(r × )
It components
∧
∧
∧
L x = y p z z p y = ih( y z ) z y ∧ ∧ ∧ ∧ ∧ L y = z p x x p z = ih( z x ) x z ∧ ∧ ∧ ∧ ∧ L z = x p y y p x = ih( x y ) y x
The first term becomes
∧2 ∧ ∧ ∧2 ∧2 ∧ ∧ ∧2
∧2
∧
∧2
∧
( L y L x - L x L y ) = L y ( L y L x ) - L x L y ) = L y (-ih Lz + L x L y ) - L x L y ) = -ih L y Lz + ( ih Lz + L x L y ) L y L x L y = -ih( L y Lz + Lz L y )
[ Li , x j ] = ihε ijk xk [ Li , p j ] = ihε ijk pk
The square of the angular momentum operator
L = Lx + L y + L z
∧2 ∧ ∧2 ∧2 ∧2 ∧
∧2
∧ 2
∧2
∧2
[L , L x ] = [L x + L y + L z , L x ] = [L y , L x ] + [L z , L x ] = (L y L x - L x L y ) + (L z L x - L x L z )
if AB + BA = 0, A and B are anticommuting (反对易) .
AB + BA = [A, B]+
operator commutator satisfy
[ A, B ] = [ B, A] [ A, B + C ] = [ A, B ] + [ A, C ] [ A, B C ] = B[ A, C ] + [ A, B ] C [ A B, C ] = A[ B, C ] + [ A, C ] B [A, [B, C]] + [B, [A, C]] + [C, [A, B]] = 0 (Jacobi恒等式)
∧2 ∧2 ∧2
(3) Kinetic operator
In Cartesian coordinate
∧ 2
h2 2 2 h2 p 2 = T= ( 2 + 2 + 2)= y 2m 2m x z 2m
∧
In polar coordinate
1 h2 1 2 h2 1 2 h2 [ 2 (r ) + 2 θ , ] = (r ) T = θ , 2 2 2m r r r r 2m r r r 2mr
∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧
∧
∧
∧
∧
∧
∧
[ x , p y ] = [ x , p z ] = [ y , p x ] = [ y , p z ] = [ z , p x ] = [ z , p y ] = 0
( xα p β p β xα ) = ihδ αβ , α , β = x,y,z
d , dx
∧
ψ ,
Lu = w
Linear operator
L(α1u1 + α 2u2 ) = α1 L u1 + α L u2
∧
∧
∧
x = x,
∧
p x = ih x
∧
linear operator nonlinear operator
α1u1 + α 2u2 ≠ α1 u1 + α 2 u2
∧ ∧ ∧ ∧ ∧
L x L y L y L x = ( y p z z p y )( z p x x p z ) ( z p x x p z )( y p z z p y ) = y ( p z z ) p x + zx p z p x yx p z p z z p y p z
2 ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧
∧
x = x, y = y , z = z
∧
∧
∧
∧
Momentum operator
p = ih
Its components
∧ ∧ p x = ih , p y = ih , p z = ih x y z
∧
Commutator of x and p
[ x , p x ] = [ y , p y ] = [ z , p z ] = ih