Using the properties of the raising and lowering operators for the 1 dimensional simple harmonic oscillator...
Problem #7 The quantum harmonic oscillator Hamiltonian, expressed in terms of raising and lowering operators, is We also know that Using these two statements, show that if then both Yn and ah have well-defined energies. Give these energies in terms of μη. Problem #7 The quantum harmonic oscillator Hamiltonian, expressed in terms of raising and lowering operators, is We also know that Using these two statements, show that if then both Yn and ah have well-defined energies. Give these energies...
3. Between the Two Methods, I Prefer the Ladder Using the raising and lowering operators, and their properties as discussed in class, work out the explicit form for the second excited state of the harmonic oscillator, that is, derive the normalized wave function pa (x) for the SHO. Show explicitly that the function you derived is orthogonal to the ground state wave function ho(x).
what is the 2-dimensional phase space density(=1-dimensional particle distribution function) of a simple harmonic oscillator? using dirac delta function.
Estimate the ground-state energy of a one-dimensional simple harmonic oscillator using (50) = e-a-l as a trial function with a to be varied. For a simple harmonic oscillator we have H + jmwºr? Recall that, for the variational method, the trial function (HO) gives an expectation value of H such that (016) > Eo, where Eo is the ground state energy. You may use: n! dH() ||= TH(c) – z[1 – H(r)], 8(2), dx S." arcade an+1 where (x) and...
2. Consider a one-dimensional simple harmonic oscillator. Do the following algebraically. 2. Consider a one -dimensional simple harmonic oscillator. Do the following algebraically. a) Construct a linear superposition of I0) and |1) by choosing appropriate phases, such that (x) is as large as possible. i) Suppose the oscillator is in the state la) att 0. Find la) and x) ii) Evaluate the expectation value(x) at t 〉 0. ii Evaluate (A x)2) att > 0. b) Answer the same questions...
Thank you in advance =)! The angular momentum raising and lowering operators are defined by on9 Although the angular momentum operators are Hermitian, the raising and lowering operators L, and L are not. Show that (a) [L,L] = 0 The angular momentum raising and lowering operators are defined by on9 Although the angular momentum operators are Hermitian, the raising and lowering operators L, and L are not. Show that (a) [L,L] = 0
Find the chemical potential of a one dimensional harmonic oscillator and a two dimensional harmonic oscillator. Please show all work. Thanks!
6 The Fermionic Oscillator Suppose that we constructed a harmonic oscillator Hamiltonian H in terms of raising and lowering operators a+,a in the usual way, such that but now whereaa obey the anticommutation relationn (Be careful! The a+,a are operators, rather than numbers.) (a) Suppose I give you a wavefunction that solves the time-independent Schrödinger equation, i.e. such that HUn-EUn-hw (n + ) ψη. Is a+Un also a solution to the time-independent Schrödinger equation If so, what is its energy...
consider a physical system 1. Consider a one-dimensional simple harmonic oscillator. a. Using +mw 2h mw ip ip mw evaluate (mlixln) (mlpln), (m+pxn) mn)(mpn b. Check that the virial theorem holds for the expectation values of the kinetic and P) the potential energy taken with respect to an energy eigenstate, i.e, the potential energy taken with respect to an energy eigenstate, 1e, V 2m 2
Consider a harmonic oscillator with Hamiltonian given by ?=(p^2/2m)+(1/2)X^2 = (a+)(a-)+(1/2) The current system state is the superposition of the lowest and next-to-lowest energy eigenstates that gives the most negative possible value for the average position, use raising and lowering operators to derive the average momentum for this state. then, simplify using ħ = ? = 1