1. Quantum harmonic oscillator (a) Derive formula for standard deviation of position measurement on a particle prepared in the ground state of harmonic oscillator. The formula will depend on h, m andw (b) Estimate order of magnitude of the standard deviation in (a) for the LIGO mirror of mass 10 kg and w 1 Hz. (c) A coherent state lo) is defined to be the eigenstate of the lowering operator with eigenvalue a, i.e. à lo)a) Write la) as where...
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...
A particle with mass m is in a one-dimensional simple harmonic oscillator potential. At time t = 0 it is described by the state where lo and l) are normalised energy eigenfunctions corresponding to energies E and Ey and b and c are real constants. (a) Find b and c so that (x) is as large as possible. b) Write down the wavefunction of this particle at a time t later c)Caleulate (x) for the particle at time t (d)...
First four harmonic oscillator normalized wavefunctions 1/4 Y.-(4)"-** 4, = 1/4 v2y ev2 1/4 Y, =| -1)ev¾2 1/4 - 3y)e¬v³½ y =ax 1. Consider a harmonic oscillator with a = 1. a) Prove that these eigenstates are all orthonormal b) Plot the first four eigenstates. How would doubling the mass change the eigenfunctions? c) Pick one eigenstate, and show that it is a solution to the Schrodinger Equation, that is, show that V? on (x) + w²ma? ¢n (x) =...
Please solve with the explanations of notations 1. The two dimensional Harmonic Oscillator has the Hamiltonian n, n'>denotes the state In> of the x-oscillator and In'> of the y-oscillator. This system is perturbed with the potential energy: Hi-Kix y. The perturbation removes the The perturbation removes the degeneracy of the states | 1,0> and |0,1> a) In first order perturbation theory find the two nondegenerate eigenstates of the full b) Find the corresponding energy eigenvalues. На Hamiltonian as normalized linear...
Consider the dimensionless harmonic oscillator Hamiltonian, (where m = h̄ = 1). Consider the orthogonal wave functions and , which are eigenfunctions of H with eigenvalues 1/2 and 5/2, respectively. with p=_ïda 2 2 We were unable to transcribe this imageY;(r) = (1-2x2)e-r2/2 (a) Let фо(x-AgVo(x) and φ2(x) = A2V2(x) and suppose that φ。(x) and φ2(x) are normalized. Find the constants Ao and A2. (b) Suppose that, at timet0, the state of the oscillator is given by Find the constant...
ONLY (e) (f) NEEDED THANK YOU :) Question 3 Consider the one-dimensional harmonic oscillator, and denote its properly normalised energy eigenstates by { | n〉, n = 0, 1, 2, 3, . . .). Define the state where α is a complex number, and C is a normalisation constant. (a) Use a Campbell-Baker-Hausdorff relation (or otherwise) to show that In other words, | α > is an eigenstate of the (non-Hermitian) lowering operator with (complex) eigenvalue α. (b) During lectures...
a 1/4 1) Show that Wo is an eigenfunction of the harmonic oscillator Schrödinger equation. 1/2 4.(x) = where a = ħ2 day 24 + ħ2 "*e-ax?12 (kg) 1 kx+]y(x) = 01 dx2 2
Consider the harmonic oscillator wave function 1/4 where α = (-)"*. Here k, is the stiffness coefficient of the oscillator and m is mass. Recall that the oscillation frequency iso,s:,k, / m In class we showed that Ψ0(x) Is an eigenfunction of the Hamiltonian, with an eigenvalue Eo (1/2)ha a) Normalize the wave function in Eq.(1) b) Graph the probability density. Note that a has units of length and measures the "width" of the wave function. It's easier to use...
4- FOR a Quartun harmonic oscillator OF MASS M, Show That The FUNCTION f(x)= x ě * 2 is EIGENFUNCTION Of The Hamiltonian. Give The genualue, Alue. x= (mk) Esln+ 1 l hv 2 -- For The 37 Excited STATE of the RiGiD ROTOR calculate the energy, the Angular momentul & Lz .