By employing the prescribed definitions of the raising and lowering operators pertaining tothe one-dimensional harmonic oscillator:x=ħ2mω-(â+ + â_)hmwê = iCompute the expectation values of the following quantities for the nth stationary staten.Keep in mind that the stationary states form an orthogonal set.2· (â+ − â_)[ pm 4ndxYmVndx = 8mna. The position of particle (x)b. The momentum of the particle (p).c. (x²)d. (p²)e. Confirm that the uncertainty principle is satisfied for all values of n

Answered: Image /qna-images/answer/d6ca5882-8922-44e6-9c77-5c2957123072.jpg

Answered: By employing the prescribed definitions…

Similar questions

PROBLEM 2 Calculate the probability distribution of momenta p for a ld oscillator in the ground state (n = 0). Calculate the mean square dispersions (x2), (p²), and the product (r2)(p²).
(c) Consider a system of two qubits with canonical basis states {|0) , |1)}. Write down an example for a two- qubit density matrix corresponding to a separable pure state and an example for a two-qubit density matrix corresponding to an entangled pure state.
Consider a particle in a one-dimensional rigid box of length a. Recall that a rigid box has U (x) = 00 for x a, and U () = 0 for 0 )
11. Evaluate (r), the expectation value of r for Y,s (assume that the operator f is defined as "multiply by coordinate r).Why does (r) not equal 0.529 for Y,,? In this problem,use 4ardr = dt.
The Hamiltonian for the one dimensional quantum oscillator is 1 p² 1 Ĥ = 1² + ½ k²² = 12 + √ mw² à² 2m 2m 2 where k = mw². 1) Define the operators ₁₁ and ₁₁ such that Ĥ = ½ħw (p² + ²). Define Ĥ2 as a function of 1 and p₁ such that Ĥ = hwĤ₂. - 2) Let us define the new operators â (1 + i₁) and â† = ½(î₁ — ip₁). Express ₁ and p₁ as a function of â and â³. Knowing that [^^1,î₁] = i and [1, 1] = -i, calculate âât and â†â. Express Ĥ2 as a function of a and at. 3) Let us define Ñ such that Ĥ₂ = Ñ + ½. Knowing that Ĥ, Ĥ₂ and Ñ have the same eigenstates, what are their corresponding eigenvalues?
Consider a finite potential step with V = V0 in the region x < 0, and V = 0 in the region x > 0 (image). For particles with energy E > V0, and coming into the system from the left, what would be the wavefunction used to describe the “transmitted” particles and the wavefunction used to describe the “reflected” particles?
O Consider the kinetic energy matrix elements between Hydrogen states (n' = 4, l', m'| |P|²| m -|n = 3, l, m), = for all the allowed l', m', l, m values. What kind of operator is the the kinetic energy (scalar or vector)? Use this to determine the following. For what choices of the four quantum numbers (l', m', l, m) can the matrix elements be nonzero (e.g. (l', m', l, m) (0, 0, 0, 0),...)? Which of these nonzero values can be related to each other (i.e. if you knew one of them, you could predict the other)? In this sense, how many independent nonzero matrix elements are there? (Note: there is no need to calculate any of these matrix elements.)
Needs Complete solution with 100 % accuracy.
Find a potential function for F or determine that F is not conservative. (If F is not conservative, enter NOT CONSERVATIVE.) F = (2xy + 3, x2 – 22, -2y)