Quantum harmonic oscillator Definition and 103 Threads

Homework Statement Given a quantum harmonic oscillator, calculate the following values: \left \langle n \right | a \left | n \right \rangle, \left \langle n \right | a^\dagger \left | n \right \rangle, \left \langle n \right | X \left | n \right \rangle, \left \langle n \right | P \left | n...

For a set of energy eigenstates |n\rangle then we have the energy eigenvalue equation \hat{H}|n\rangle = E_{n}|n\rangle. We also have a commutator equation [\hat{H}, \hat{a^\dagger}] = \hbar\omega\hat{a}^{\dagger} From this we have \hat{a}^{\dagger}\hat{H}|n\rangle =...

Homework Statement The question is from Sakurai 2nd edition, problem 3.21. (See attachments) ******* EDIT: Oops! Forgot to attach file! It should be there now.. *******The Attempt at a Solution Part a, I feel like I can do without too much of a problem, just re-write L as L=xp and then...

Homework Statement Compute ##\left \langle x^2 \right\rangle## for the states ##\psi _0## and ##\psi _1## by explicit integration. Homework Equations ##\xi\equiv \sqrt{\frac{m \omega}{\hbar}}x## ##α \equiv (\frac{m \omega}{\pi \hbar})^{1/4}## ##\psi _0 = α e^{\frac{\xi ^2}{2}}##The Attempt at...

This is more of a conceptual question and I have not had the knowledge to solve it. We're given a modified quantum harmonic oscillator. Its hamiltonian is H=\frac{P^{2}}{2m}+V(x) where V(x)=\frac{1}{2}m\omega^{2}x^{2} for x\geq0 and V(x)=\infty otherwise. I'm asked to justify in...

I was reading through my Principles of Quantum Mechanics textbook and arrived at the section that discusses the quantum harmonic oscillator. In this discussion the equation ψ"-(y^2)ψ=0 presents itself and a solution is given as ψ=(y^m)*e^((-y^2)/2), similar to a gaussian function i assume. My...

Homework Statement Consider an electron confined by a 1 dimensional harmonic potential given by ## V(x) = \dfrac{1}{2} m \omega^2 x^2##. At time t=0 the electron is prepared in the state \Psi (x,0) = \dfrac{1}{\sqrt{2}} \psi_0 (x) + \dfrac{1}{\sqrt{2}} \psi_4 (x) with ## \psi_n (x) = \left(...

The Wigner function, W(x,p)\equiv\frac{1}{\pi\hbar}\int_{-\infty}^{\infty} \psi^*(x+y)\psi(x-y)e^{2ipy/\hbar}\, dy\; , of the quantum harmonic oscillator eigenstates is given by, W(x,p) = \frac{1}{\pi\hbar}\exp(-2\epsilon)(-1)^nL_n(4\epsilon)\; , where \epsilon =...

Homework Statement Find the uncertainty of the kinetic energy of a quantum harmonic oscillator in the ground state, using \left\langle p^2_x \right\rangle = \displaystyle\frac{\hbar^2}{2a^2} and \left\langle p^4_x \right\rangle = \displaystyle\frac{3\hbar^2}{4a^2} Homework Equations...

Homework Statement What is the effect of the sequence of ladder operators acting on the ground eigenfunction \psi_0 Homework Equations \hat{A}^\dagger\hat{A}\hat{A}\hat{A}^\dagger\psi_0The Attempt at a Solution I'm not sure if I'm right but wouldn't this sequence of opperators on the ground...

Homework Statement I am unsure as to a step in Griffiths's derivation of the quantum harmonic oscillator. In particular, I am wondering how he arrived at the equations at the top of the second attached photo, from the last equation (at the bottom) of the first photo (which is the recursion...

Homework Statement Show that the following is an eigenfunction of \hat{H}_{QHO} and hence find the corresponding eigenvalue: u(q)=A (1-2q^2) e^\frac{-q^2} {2} Homework Equations Hamiltonian for 1D QHO of mass m \hat{H}_{QHO} = \frac{\hat{p}^2}{2m} + \frac{1}{2} m \omega^2 x^2...

Homework Statement Particle of mass m undergoes simple harmonic motion along the x axis Normalised eigenfunctions of the particle correspond to the energy levels E_n = (n+ 1/2)\hbar\omega\ \ \ \ (n=0,1,2,3...) For the two lowest energy levels the eigenfunctions expressed in natural...

Hi All, If there is something fundamentally wrong in my understanding of quantum mechanics, pardon me for I have just started learning it. We know that if we can come up with a solution for Schrodinger Equation of a Harmonic Oscillator, then we can generate further solutions by acting on it...

Homework Statement (See attachment) Homework Equations x = \sqrt{\frac{\hbar}{2m \omega}} ( a + a^{\dagger} ) x = i \sqrt{\frac{\hbar m \omega}{2}} ( a^{\dagger} - a ) The Attempt at a Solution In part a) I was able to construct a separable Hamiltonian for the harmonic...

I need to find the value σ for which: ψ0(x) = (2πσ)-1/4 exp(-x2/4σ) is a solution for the Schrodinger equation I know the equation for the QHO is: Eψ = (P2/2m)ψ + 1/2*mw2x2ψ I've tried normalizing the wavefunction but I end up with a σ/σ term :( Any help would be greatly...

Over which interval do the wave functions of a harmonic oscillator form a complete and orthogonal system? Is it (-inf,+inf)? The case with particle in a box is rather clear(system is complete and orthogonal only for the interval of the well), however the harmonic oscillator is a bit less intuitive.

Homework Statement Consider a quantum mechanical particle moving in a potential V(x) = 1/2mω2x2. When this particle is in the state of lowest energy, A: it has zero energy B: is located at x = 0 C: has a vanishing wavefunction D: none of the above Homework Equations The...

Homework Statement I need to transform the Hamiltonian of a coupled Harmonic Oscillator into the sum of two decoupled Hamiltonians (non-interacting oscillators). Homework Equations H = H1 + H2 + qxy, where H1=0.5*m*omega^2*x^2+0.5m^-1P_x^2 and H2=0.5*m*omega^2*y^2+0.5m^-1P_y^2, and q is...

At classical harmonic oscillator, total energy is proportional to square of frequency, but at quantum harmonic oscillator, total energy is proportional to frequency. Are those two frequencies the same? How it is with transition from quantum harmonic oscillator to classical harmonic oscillator...

Hey guys, For a particular problem I have to determine the total degeneracy across N 3-D Quantum Harmonic oscillators. Given that the degree of degeneracy for a 3-D harmonic oscillator is given by: (n+1)(n+2)/2 and the Total energy of N 3d quantum harmonic oscillators is given by...

Hello everybody, recently in my quantum mechanical course we were introduced to the concept of the quantum harmonic oscillator. My question is: is there a physical significance attached to the fact that the classical turning points overlap with the sign change of the second derivative of the...

Hi, so i am looking at the quantization of the harmonic oscillator and i have the following equation... ψ''+ (2ε-y^{2})ψ=0 I am letting y\rightarrow \infty to get... ψ''- y^{2}ψ=0 It says the solution to this equation in the same limit is... ψ= Ay^{m}e^{\pm y^{2}/2} The positive...

Hey guys I was just looking over a past homework problem and found something I'm not too sure on - A particle is in the ground state of a Harmonic potential V (x) = 0.5mω2x2 If you measured the energy, what are the possible results, and with what probabilities? Now I know the answer...

Homework Statement Consider the Hamiltonian H=\frac{p^2}{2M}+\frac{1}{2}\omega^2r^2-\omega_z L_z Determine its eigenstates and energies. 2. The attempt at a solution I want to check my comprehension; by eigenstate they mean \psi(r) from the good old H\psi(r)=E\psi(r) and...

Consider the usual 1D quantum harmonic oscillator with the typical hamiltonian in P and X and with the usual ladder operators defined. i) I have to prove that given a generic wave function \psi , \partial_t < \psi (t) |a| \psi (t)> is proportional to < \psi (t) | a | \psi (t) > and...

Homework Statement A particle of mass m is placed in the ground state of a one-dimensional harmonic oscillator potential of the form V(x)=1/2 kx2 where the stiffness constant k can be varied externally. The ground state wavefunction has the form ψ(x)\propto exp(−ax2\sqrt{k}) where a...

Homework Statement What are the stationary states of an isotropic 3D quantum harmonic oscillator in a potential U(x,y,z) = {1\over2}m\omega^2 (x^2+y^2+z^2) in the form \psi(x,y,z)=f(x)g(y)h(z) and how many linearly independent states have energy E=({3\over 2}+n)\hbar\omega? Homework...

Homework Statement Let's consider a harmonic oscillator with a harmonic perturbation: H = \frac{p^2}{2} + \frac{x^2}{2} + a \frac{x^2}{2}. Exact solution is known, but we want to derive it using perturbation theory. More specifically, suppose we want to obtain a series for the ground state...

Homework Statement A particle is in the ground state of a harmonic oscillator with classical frequency w. Suddenly the classical frequency doubles, w -> w' = 2w without initially changing the wavefunction. Instantaneously afterwards, what is the probability that a measurement of energy...

Homework Statement Given that a^+|n\rangle=\sqrt{n+1}|n+1\rangle a|n\rangle=\sqrt{n}|n-1\rangle and that the other eigenstates |n> are given by |n\rangle=\frac{(a^+)^n}{\sqrt{n!}}|0\rangle where |0> is the lowest eigenstate. Define for each complex number z the coherent state...

Homework Statement For a particle of mass m moving in the potential V(x) = \frac{1}{2}m\omega^2x^2 (i.e. a harmonic oscillator), it is often convenient to express the position and momentum operators in terms of the ladder operators a_{\pm}: x = \sqrt{\frac{\hbar}{2m\omega}}(a_+ + a_-) p =...

A classical harmonic oscillator follows a smooth, sinusoidal path of oscillation. Since on a quantum level energy levels are discrete, does a quantum harmonic oscillator actually oscillate in the everyday sense?

Hello. I am trying to use the following equation: a\left|\psi_n\right\rangle=\sqrt{n}\left|\psi_{n-1}\right\rangle (where a is the "ladder operator"). What happens when I substitute \psi_n with \psi_0?

Quantum Harmonic Oscillator Operator Commution (solved) EDIT This was solved thanks to CompuChip! The entire post is also not very interesting as it was a basic mistake :P No need to waste time This is not homework (I am not currently in college :P), but it is a mathematical question I'm...

Ok, so I am trying understand how to derive the following version of the Schrodinger Equation for QHO: \frac{d^2u}{dz^2} + (2\epsilon-z^2)u=0 where \ 1. z=(\frac{m\omega}{hbar})^{1/2}x and \ 2. \epsilon= \frac{E}{hbar\omega} I've started with the TISE, used a potential of...

Homework Statement Consider a harmonic oscillator with mass=0.1kg, k=50N/m , h-bar=1.055x10-34 Let this oscillator have the same energy as a mass on a spring, with the same k and m, released from rest at a displacement of 5.00 cm from equilibrium. What is the quantum number n of the state of...

Find the time dependence of the expectation value <x> in a quantum harmonic oscillator, where the potential is given by V=\frac{1}{2}kx^2 I'm assuming some wavefunction of the form \Psi(x,t)=\psi(x) e^{-iEt/\hbar} When I apply the position operator, I get: <x>=\int_{-\infty}^\infty...

Homework Statement A quantum mechanical harmonic oscillator with resonance frequency ω is placed in an environment at temperature T. Its mean excitation energy (above the ground state energy) is 0.3ħω. Determine the temperature of this system in units of its Einstein-temperature ΘE = ħω/kB...

Hello everybody, I noticed these questions are lengthy. If you want to skip my introduction, just scroll down to the questions. I put *** next to each one. I just started Quantum Theory I this semester and I have a question (actually two questions) regarding the quantum harmonic...

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc4.html#c1 I know the energy should be E = \frac{{{p^2}}}{{2m}} + \frac{1}{2}m{\omega ^2}{x^2} But I can't figure out why the minimum energy is related to \Delta p

I've followed this: http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc3.html#c1, up to the part where it gets to here: . The guide says: "Then setting the constant terms equal gives the energy"? Am I being stupid? I really can't see where that equations come from.

Homework Statement I have to find the minimum and maximum values of the uncertainty of \Deltax and specify the times after t=0 when these uncertainties apply. Homework Equations The wave function is Ψ(x, 0) = (1/√2) (ψ1(x)+ iψ3(x)) and for all t is Ψ(x, t) = (1/√2)...

Hi! Would anyone be able to point me toward a detailed explanation of determining the Hamiltonian of a polyatomic quantum oscillator? My current text does not explain the change of coordinates ("using normal coordinates or normal modes") in detail. All I can find is material on a diatomic...

Can someone tell me if there is a difference in the moving motion between a quantum harmonic oscillator and a simple harmoic oscillator. Also, does anoyone know a good site where i could learn more on quantum harmonic oscillator. ty

The question is as follows: Suppose that, in a particular oscillator, the angular frequency w is so large that its kinetic energy is comparable to mc2. Obtain the relativistic expression for the energy, En of the state of quantum number n. I don't know how to begin solving this question. I...

Does anyone know why a harmonic potential gives rise to coherent states? In other words, what is special about a quadratic potential that causes the shifted ground state to oscillate like a classical particle without dispersing so as to saturate the uncertainty principle? Any help or insight...

I am working with the following harmonic oscillator formula. \psi_n \left( y \right) = \left( \frac {\alpha}{{\pi}} \right) ^ \frac{1}{{4}} \frac{1}{{\sqrt{2^nn!}}}H_n\left(y\right)e^{\frac{-y^2}{{2}}} Where y = \sqrt{\alpha} x And \alpha = \frac{m\omega}{{\hbar}} I...

Homework Statement Is there any way to find <\varphi_{n}(x)|x|\varphi_{m}(x)|> (where phi_n(x) , phi_m(x) are eigenfunction of harmonic oscillator) without doing integral ? Homework Equations perhaps orthonormality of hermite polynomials ...

looking at the quantum mechanical harmonic oscillator, one has the differential equation in the form: \frac{d^2\psi}{du^2}+(\alpha-u^2)\psi=0 when a person who doesn't know any physics sees the equation, he will try a serial solution for psi, and he will find a solution with some recursive...