4. Prove that if there are two operators A, ˆ Bˆ, such that their commutator [A,...

4. Prove that if there are two operators A, ˆ Bˆ, such that their commutator [A, ˆ Bˆ] = λ where
λ is a constant. Show that (6 points)

exp[μ(Aˆ + Bˆ)] = exp(μAˆ) exp(μBˆ) exp(−μ
2λ/2)

where given an operator Xˆ; exp(tXˆ) = 1 + tXˆ +
1
2
t
2Xˆ 2 + ... +
1
n!
t
nXˆ n + ... (the usual

exponential series implemented for operators).

Expert Solution

genius_generous answered 6 months ago

Prove that for a wave function to be a simultaneous eigenfunction of two operators A and...

Prove that for a wave function to be a simultaneous eigenfunction of two operators A and C, the operators must commute.

2. (a). Prove that the sum of the two projection operators is not a projection operator,...

2. (a). Prove that the sum of the two projection operators is not a projection operator, unless the multiplication of the two projection operators produces a zero value. (b). Prove that the result of the multiplication of two projection operators is not a projection operator, unless the two projection operators are hermitian.

First prove A and B to be Hermitian Operators, and then prove (A+B)^2 to be hermitian...

First prove A and B to be Hermitian Operators, and then prove (A+B)^2 to be hermitian operators.

prove that positive operators have unique positive square root

Suppose V is finite-dimensional and S, T are operators on V . Prove that ST is...

Suppose V is finite-dimensional and S, T are operators on V . Prove that ST is bijective if and only if S and T are both bijective. Note: Don’t forget that bijective maps are precisely those that have an inverse!

Prove that R(4; 4) = 18.

Show that the tensor product of two positive operators is positive.

Mass mA =2 .1 kg traveling with velocity~ vA = (1 .2m /s)ˆ x(2.3m /s)ˆ y...

Mass mA =2 .1 kg traveling with velocity~ vA = (1 .2m /s)ˆ x(2.3m /s)ˆ y collides with mass mB =4 .2 kg at rest. After collision, mass mA leaves with velocity |~ v0A| =0 .50 m/s at an angle = 35 . (a) Calculate the total initial momentum of the system before the collision. (b) What is the momentum ~ p 0 A of mass mA after the collision. Give your answer in ˆ x and ˆ y component...

Vector operators

Prove that any two groups with one element are isomorphic. Prove that any two groups with...

Prove that any two groups with one element are isomorphic. Prove that any two groups with two elements are isomorphic. Prove that any two groups with three elements are isomorphic.

Question