Question

A particle in a simple harmonic oscillator potential V (x) = 1 /2mω^2x^2 has an initial...

A particle in a simple harmonic oscillator potential V (x) = 1 /2mω^2x^2 has an initial wave function
Ψ(x,0) = (1/ √10)(3ψ1(x) + ψ2(x)) ,
where ψ1 and ψ2 are the stationary state solutions of the first and second energy level. Using raising and lowering operators (no explicit integrals except for orthonomality integrals!) find <x> and <p> at t = 0

Science   Advanced-Physics   
0 0
Add a commentTranscribed image text
Next >

Homework Answers

Answer #1

0 0
Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Log In

Sign Up

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
A particle in a strange potential well has the following two lowest-energy stationary states: ψ1(x) =...
A particle in a strange potential well has the following two lowest-energy stationary states: ψ1(x) = C1 sin^2 (πx) for − 1 ≤ x ≤ 1 ψ2(x) = C2 sin^2 (πx) for − 1 ≤ x ≤ 0 ψ2(x) = −C2 sin^2 (πx) for 0 ≤ x ≤ 1 The energy of ψ1 state is ¯hω. The energy of ψ2 state is 2¯hω. The particle at t = 0 is in a superposition state Ψ(x, t = 0) = 1/√...
For a particle in the first excited state of harmonic oscillator potential, a) Calculate 〈?〉1, 〈?〉1,...
For a particle in the first excited state of harmonic oscillator potential, a) Calculate 〈?〉1, 〈?〉1, 〈? 2〉1, 〈? 2〉1. b) Calculate (∆?)1 and (∆?)1. c) Check the uncertainty principle for this state. d) Estimate the length of the interval about x=0 which corresponds to the classically allowed domain for the first excited state of harmonic oscillator. e) Using the result of part (d), show that position uncertainty you get in part (b) is comparable to the classical range of...
particle of mass m is moving in a one-dimensional potential V (x) such that ⎧ ⎨...
particle of mass m is moving in a one-dimensional potential V (x) such that ⎧ ⎨ mω2 x2 ifx>0 V (x) = 2 ⎩ +∞ if x ≤ 0 (a) Consider the motion classically. What is the period of motion in such potential and the corresponding cyclic frequency? (b) Consider the motion in quantum mechanics and show that the wave functions of the levels in this potential should coincide with some of the levels of a simple oscillator with the...
Imagine a harmonic oscillator with Hamiltonian H=p^2/2m+½x^2 For simplicity, we will assume that m=ℏ=1. First, we...
Imagine a harmonic oscillator with Hamiltonian H=p^2/2m+½x^2 For simplicity, we will assume that m=ℏ=1. First, we set up our system in the first excited state of this Hamiltonian. Second, we turn on an extra potential, V_ex(x)=x^4. Third, before the added potential has a chance to change the system state, we measure the energy.By expressing H and V_ex in terms of the raising and lowering operators, evaluate the average energy we would see if we repeated this whole process many times.
A simple harmonic oscillator has a frequency of 11.1 Hz. It is oscillating along x, where...
A simple harmonic oscillator has a frequency of 11.1 Hz. It is oscillating along x, where x(t) = A cos(ωt + δ). You are given the velocity at two moments: v(t=0) = 1.9 cm/s and v(t=.1) = -18.1 cm/s. 1) Calculate A. 2) Calculate δ.
To generate the excited states for the quantum harmonic oscillator, one repeatedly applies the raising operator...
To generate the excited states for the quantum harmonic oscillator, one repeatedly applies the raising operator ˆa+ to the ground state, increasing the energy by ~ω with each step: ψn = An(ˆa+) nψ0(x) with En = (n + 1 2 )~ω where An is the normalization constant and aˆ± ≡ 1 √ 2~mω (∓ipˆ+ mωxˆ). Given that the normalized ground state wave function is ψ0(x) = mω π~ 1/4 e − mω 2~ x 2 , show that the first...
1. Consider a particle with mass m in a one dimensional potential V (x) = A/x...
1. Consider a particle with mass m in a one dimensional potential V (x) = A/x + Bx, x > 0. (a) Expand the potential V about its minimum value to quadratic order. (b) Find the lowest two stationary state energies in terms of A, B, m and fundamental constants.
Consider an electron bound in a three dimensional simple harmonic oscillator potential in the n=1 state....
Consider an electron bound in a three dimensional simple harmonic oscillator potential in the n=1 state. Recall that the e- has spin 1/2 and that the n=1 level of the oscillator has l =1. Thus, there are six states {|n=1, l=1, ml, ms} with ml= +1, 0, -1 and ms = +/- 1/2. - Using these states as a basis find the six states with definite j and mj where J = L +s - What are the energy levels...
5. The wavefunction of a particle is ?(?) = ??−??/2 for x>0 and ?(?) = ????/2...
5. The wavefunction of a particle is ?(?) = ??−??/2 for x>0 and ?(?) = ????/2 for x<0. Find the corresponding potential energy, constant A, and energy eigenvalue. 6. The hydrogen molecule H2 can be treated as a vibrating system (simple harmonic oscillator), with an effective force constant ? = 3.5 × 10^3 eV/nm2. Compute the zero-point (ground state) energy of one of the protons in H2. How does it compare with the molecular binding energy of 4.5 eV? Compute...
In this problem we are interested in the time-evolution of the states in the infinite square...
In this problem we are interested in the time-evolution of the states in the infinite square potential well. The time-independent stationary state wave functions are denoted as ψn(x) (n = 1, 2, . . .). (a) We know that the probability distribution for the particle in a stationary state is time-independent. Let us now prepare, at time t = 0, our system in a non-stationary state Ψ(x, 0) = (1/√( 2)) (ψ1(x) + ψ2(x)). Study the time-evolution of the probability...
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT