For the nth stationary state of the harmonic oscillator, using the algebraic method, show that: = ( 2 ) mw 3(2n²+2n+1),
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- Be *(1) the position operator for a particle subjected to a potential of a one-dimensional harmonic oscillator P mox (Ĥ =+ 2m 2 Evaluate [î(t),î(0)] Heisenberg's chart inShow that ? (x,t) = A cos (kx - ?t) is not a solution to the time-dependent Schroedinger equation for a free particle [U(x) = 0].Using the eigenvectors of the quantum harmonic oscillator Hamiltonian, i.e., n), find the matrix element (6|X² P|7).
- A neutron of mass m of energy E a,V(x) = Vo ) II. Estimate the kinetic energy of the neutron when they reach region III.Consider a weakly anharmonic a 1D oscillator with the poten- tial energy m U(x) = w?a² + Ba* 2 Calculate the energy levels in the first order in the small anharmonicity parameter 3 using TIPT and the ladder operators.consider an infinite square well with sides at x= -L/2 and x = L/2 (centered at the origin). Then the potential energy is 0 for [x] L/2 Let E be the total energy of the particle. =0 (a) Solve the one-dimensional time-independent Schrodinger equation to find y(x) in each region. (b) Apply the boundary condition that must be continuous. (c) Apply the normalization condition. (d) Find the allowed values of E. (e) Sketch w(x) for the three lowest energy states. (f) Compare your results for (d) and (e) to the infinite square well (with sides at x=0 and x=L)