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.

Expert Solution

In classical mechanics, the movement of a particle (or system of particles) is completely determined by the Lagrangian or equivalently the Hamiltonian , a function of the generalized coordinates q, generalized velocities and its conjugate momenta:

If either L or H are independent of a generalized coordinate q, meaning the L and H do not change when q is changed, which in turn means the dynamics of the particle are still the same even when q changes, the corresponding momenta conjugate to those coordinates will be conserved (this is part of Noether's theorem, and the invariance of motion with respect to the coordinate q is a symmetry). Operators in classical mechanics are related to these symmetries.

More technically, when H is invariant under the action of a certain group of transformations G:

.

the elements of G are physical operators, which map physical states among themselves.

You can find through this.

queen_honey_blossom answered 9 months ago

In a simultaneous pricing game there are two operators. If one of them prices low, he...

In a simultaneous pricing game there are two operators. If one of them prices low, he gets all the customers, a payoff of 12, while other gets zero. If both price high they each get a payoff of 6 and if both price low, they get a payoff of 5. What is the best response of Operator 2 to Operator 1 pricing high? Select one: a. Leave the market b. Also price high c. Price low d. Price at the...

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).

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.

Prove that even number of fermions produces boson using wave function.

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 (a) that ψ ± = N (x ± iy)f(r) is an eigenfunction of L^2 and...

Prove (a) that ψ ± = N (x ± iy)f(r) is an eigenfunction of L^2 and Lz and set the eigenvalues corresponding. (b) Construct a wave function ψ_0(r) that is an eigenfunction of L^2 whose eigenvalue is the same as that of a), but whose eigenvalue Lz differs by a unit of the one found in a). (c) Find an eigenfunction of L^2 and Lx, analogous to those of parts a) and b), which have the same eigenvalue L^2 but...

prove that positive operators have unique positive square root

Using identities for time derivatives and del operators, show that the solution of the inhomogeneous wave...

Using identities for time derivatives and del operators, show that the solution of the inhomogeneous wave equation for the potential V given by the Lorentz gauge (the expression containing an integral over volume), does indeed satisfy the inhomogeneous wave equation.

How the wave function and wave packets can be used to explain the particle-wave properties of...

How the wave function and wave packets can be used to explain the particle-wave properties of electrons?

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!

Question