Let the quantum state be y(x,y,z) = zf(r) + z?g(r) Write it as a linear combination of the eigenstates of operators: and lz
Q: Consider a quantum system in the initial state ly (0) = |x,) at r = 0, and the Hamiltonian H = (252…
A: Given:Initial state; ∣ψ(0)⟩=∣x+⟩Hamiltonian; H=(2ℏΩ00ℏΩ)Constant frequency; Ω We can express the…
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A: Using property of angular momentum operator we can solve the problem as solved below
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Q: The z-direction angular momentum operator in quantum mechanics is given as (SPH eqn 44): Ә L3=-ih де…
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Q: 4. Consider an operator  satisfying the commutation relation [Â, †] = 1. (a) Evaluate the…
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Q: Starting from the definition of the partition function, Z = Ei e-Bei, prove the following: a) (E): =…
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Q: Consider a system with Hamiltonian operator H that is in a state k with energy Ek, where Ĥ WK = Ex…
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Q: PROBLEM 2 Calculate the probability distribution of momenta p for a ld oscillator in the ground…
A: Solution: The ground state is n =0. The position and momentum operator in terms of raising and…
Q: (a) Consider the following wave function of Quantum harmonic oscillator: 3 4 V(x, t) =Vo(x)e¯REot…
A: a) From question So expectation value of x will be, {*since wave function of ground and exited…
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- By taking the derivative of the first equation with respect to b, show that the second equation is true. Use this result to determine △x and △p for the ground state of the simple harmonic oscialltor.A 3-D particle in a box has dimensions of x = 0 to a, y = 0 to a, and z = 0 to 2a. The normalized Wavefunction is: 7. 4 Y(x, y,z)=. n,Ty sin sin 2a sin a The Hamiltonian operator is: Ĥ = - 2m Ox a) Apply A to Y and find E, b) Show that the sets of quantum numbers(n, ny, nz) = (1,2,2), (2,1,2), and (1,1,4) represent a set of triply-degenerate energy levels.(AA) ²( ▲ B) ²≥ ½ (i[ÂÂ])² If [ÂÂ]=iñ, and  and represent Hermitian operators corresponding to observable properties, what is the minimum value that AA AB can have? Report your answer as a decimal number with three significant figures.
- 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.)Suppose that the wave function for a system can be written as 4(x) = √3 4 · Φι(x) + V3 2√₂ $2(x) + 2 + √3i 4 $3(x) and that 1(x), 2(x), and 3(x) are orthonormal eigenfunc- tions of the operator Ekinetic with eigenvalues E₁, 2E₁, and 4E₁, respectively. a. Verify that (x) is normalized. b. What are the possible values that you could obtain in measuring the kinetic energy on identically prepared systems? c. What is the probability of measuring each of these eigenvalues? d. What is the average value of Ekinetic that you would obtain from a large number of measurements?The operator în · ở measures spin in the direction of unit vector f = (nx, Ny, N₂) nx = sin cosp ny = sinesino nz = cose in spherical polar coordinates, and ở = (x, y, z) for Pauli spin matrices. (a) Determine the two eigenvalues of û.o.