Download Hermitian and unitary operator

Hermitian and unitary operator
An operator is Hermitian if it is self-adjoint:
A+ = A
Or equivalently: < ψ|A| φ> = (<φ|A|ψ>)* and so 〈A〉= < ψ|A|ψ> is real.
An operator is skew-Hermitian if B+ = -B and 〈B〉= < ψ|B|ψ> is imaginary.
In quantum mechanics, the expectation of any physical quantity has to
be real and hence an operator corresponds to a physical observable
must be Hermitian. For example, momentum operator and
Hamiltonian are Hermitian.
An operator is Unitary if its inverse equal to its adjoints:
U-1 = U+
or
UU+ = U+U = I
In quantum mechanics, unitary operator is used for change of basis.
Commutators
Operators do not commute.
Commutator: [A, B] = AB-BA
Anti-commutator: {A, B} = AB+BA
Algebra of commutators:
1. Antisymmetric: [A, B] = -[B, A]
2. Bilinear: [αA+βB+γC+…. , X] = α [A,X]+ β[B,X]+ γ[C,X]+….
[X, αA+βB+γC+…. ] = α[X, A]+ β[X, B]+ γ[X, C]+….
3. Adjoint: [A, B]+ = [B+, A+]
4. Distributive: [A, BC] = [A,B]C+B[A,C]
[BC, A] = [B,A]C+B[C,A]
5.
n −1
[A, B ] = ∑ B j[A, B]A n − j−1
n
j= 0
n −1
[A , B] = ∑ A n − j−1[A, B]B j
n
j= 0
6. Jacobi identity: [A, [B, C]] + [B, [C, A]] + [ C, [A, B]] = 0
7. If A and B are Hermitian, [A, B] is skew-Hermitian and {A, B} is Hermitian.
8. Note that [A, B]=0 and [B, C]=0 does NOT imply [B, C]=0.
Function of operators
If f(A) is a “function” of A. We can Taylor expand f(A) in a power series of A:
∞
f(A) = ∑ a n A n
n =0
∞
∴[f(A)] = ∑ a n (A + ) n = f * (A + )
+
n =0
For example,
∞
e
aA
*
=∑
n =0
a nAn
1
1
= I + aA + a 2 A 2 + a 3 A 3 + L
n!
2
6
(e A ) + = e A
+
and (eiA ) + = e -iA
+
Commutators involving function of operators :
1. [A, f(A)] = 0
but e A e B = e A + Be[A,B]/2
1
1
3. e -A Be A = B + [A, B] + [A, [A, B]] + [A,[A, [A, B]]] + L
2!
3!
2. e e ≠ e
A B
A+B
Commutators and uncertainty relation
Define operator
δA=A- 〈A〉 where 〈A〉=<ψ|A|ψ> and A2 =<ψ|A2| ψ>
Let δA|ψ>=|χ> and δB |ψ>=|φ>. Note that if A and B are Hermitian, so are δA and δB.
δA=δA+ ⇒ 〈δA2〉 = 〈δAδA〉 =<ψ|δAδA|ψ> =<ψ|δA+δA|ψ> =<χ|χ> ⇒ 〈δA2〉 = <χ|χ>
Similarly, 〈δB2〉= <φ |φ> and 〈δAδB〉 = <χ| φ>
Schwarz inequality: <χ|χ> <φ |φ> ≥ | <χ| φ>|2 ⇒ 〈δA2〉〈δB2〉 ≥ 〈δAδB〉2
Now,
δAδB =
1
2
(δAδB - δBδA) + 12 (δAδB + δBδA)
[δA, δB] + 12 {δA, δB}
=
1
2
=
[A, B] + 12 {δA, δB}
1
424
3 14243
1
2
Re al
Im aginary
∴ δAδB
2
= 14 [A, B]
δA 2 δB2 ≥ δAδB
2
2
+
1
2
{δA, δB}
⇒ δA 2 δB 2 ≥
δA 2 =
(A -
∴ ∆A 2 ∆B2 ≥ 14 [A, B]
A
2
1
4
[A, B]
δA 2 δB2 ≥
⇒
Define ∆A =
2
)
2
⇒
=
A2 - A
1
4
2
2
+
[A, B]
1
2
{δA, δB}
2
and ∆B =
∆A∆B ≥ 12 [A, B]
δ B2
2