What is Quantum harmonic oscillator: Definition and 108 Discussions
The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary smooth potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics. Furthermore, it is one of the few quantum-mechanical systems for which an exact, analytical solution is known.
Using the ladder operators I can easily compute ##E = \langle H\rangle = \hbar \omega n##, so I can find the amplitude of the classical oscillator, as ##E = \frac{1}{2} m \omega^2 x_{max}^2##, thus, ##x_{max} = \sqrt{\dfrac{2 E}{m \omega^2}} = \sqrt{\dfrac{2\hbar n}{m \omega}}##.
The...
Hello there, I am trying to solve the above and I'm thinking that the solutions will be Hermite polynomials multiplied by a decaying exponential, much like the standard harmonic oscillator problem. The new Hamiltonian would be like so:
$$H = - \frac \hbar {2m} \frac {d^2}{dx^2}\psi + \frac...
##x## can be discretized as ##x \rightarrow x_k ## such that ##x_{k + 1} = x_k + dx## with a positive integer ##k##. Throughout we may assume that ##dx## is finite, albeit tiny.
By applying the Taylor expansion of the wavefunction ##\psi_n(x_{k+1})## and ##\psi_n(x_{k-1})##, we can quickly...
I am getting that we have to operate the given Hamiltonian on the given state |α>. But what is confusing me is that since this H contains position and momentum operators which just involve variable x and partial derivative, how do I operate this H on the given α, since it seems like α is...
I began this solution by assuming a = x+iy since a is a complex number.
So I wrote expressions of <a| and |a> in which |n><n| = I.
I got the following integral:
Σ 1/πn! ∫∫ dx dy exp[-(x^2 + y^2)] (x^2 + y^2)^n I
I tried solving it using Integration by Parts but got stuck in the (x^2 + y^2)^n...
In the simple harmonic oscillator, I was told to use the raising and lowering operator to generate the excited states from the ground state. However, I am just thinking that how do we confirm that the raising operator doesn't miss some states in between.
For example, I can define a raising...
What I have tried is a completing square in the Hamiltonian so that
$$\hat{H} = \frac{\hat{p}^2}{2} + \frac{(\hat{q}+\alpha(t))^2}{2} - \frac{(\alpha(t))^2}{2}$$
I treat ##t## is just a parameter and then I can construct the eigenfunctions and the energy eigenvalues by just referring to a...
hi guys
i am trying to solve the Asymptotic differential equation of the Quantum Harmonic oscillator using power series method and i am kinda stuck :
$$y'' = (x^{2}-ε)y$$
the asymptotic equation becomes :
$$y'' ≈ x^{2}y$$
using the power series method ##y(x) = \sum_{0}^{∞} a_{n}x^{n}## , this...
For the off-diagonal term, it is obvious that (p^2+q^2) returns 0 in the integration (##<m|p^2+q^2|n> = E<m|n> = 0##). However, (pq+qp) seems to give a complicated expression because of the complicated wavefunctions of a quantum harmonic oscillator. I wonder whether there is a good method to...
Consider two harmonic oscillators, described by annihilation operators a and b, both initially in the vacuum state. Let us imagine that there is a coupling mechanism governed by the Hamiltonian H=P_A P_B, where P_i is the momentum operator for the oscillator i. For example P_A =...
Suppose we have a Hamiltonian containing a term of the form
where ∂=d/dr and A(r) is a real function. I would like to study this with harmonic oscillator ladder operators. The naïve approach is to use
where I have set ħ=1 so that
This term is Hermitian because r and p both are.*...
##\newcommand{\ket}[1]{|#1\rangle}##
##\newcommand{\bra}[1]{\langle#1|}##
I have a momentum-shifting operator ##e^{i\Delta p x/\hbar}## acting on the ground state ##\ket{0}## of the QHO, and I want to compute the overlap of this state with the n##^{th}## excited QHO state ##\ket{n}##. Given...
In the paper below I've seen a new method to solve the quantum harmonic oscillator
Introduction to the Spectrum of N=4 SYM and the Quantum Spectral Curve
It is done using the concept of quasi momentum defined as
$$p = - i \frac{d(\log \psi)}{dx}$$
See pg 7,8
Is this well know? is it discussed...
Dear PF community, I am back with a question :)
The solutions for the quantum harmonic oscillator can be found by solving the Schrödinger's equation with:
Hψ = -hbar/2m d²/dx² ψ + ½mω²x² ψ = Eψ
Solving the differential equation with ψ=C exp(-αx²/2)
gives:
-hbar/2m (-α + α²x²)ψ + ½mω²x²ψ = Eψ...
The wavefunction is Ψ(x,t) ----> Ψ(λx,t)
What are the effects on <T> (av Kinetic energy) and V (potential energy) in terms of λ?
From ## \frac {h^2}{2m} \frac {\partial^2\psi(x,t)}{\partial x^2} + V(x,t)\psi(x,t)=E\psi(x,t) ##
if we replace x by ## \lambda x ## then it becomes ## \frac...
I'm getting confused by the perturbation theory aspect of problem 2.2 in this book. We have to show that the energy eigenvalues are given by
$$E_n = \left(n + \frac{1}{2}\right) \hbar \omega + \frac{3\lambda}{4} \left(\frac{\hbar}{m\omega}\right)^2 (2n^2 + 2n + 1)$$
For the Hamiltonian...
I was reviewing the harmonic oscillator with Sakurai. Using the annihilation and the creation operators ##a## and ##a^{\dagger}##, and the number operator ##N = a^{\dagger}a##, with ##N |n \rangle = n | n \rangle##, he showed that ##a | n \rangle## is an eigenstate of ##N## with eigenvalue ##n -...
Homework Statement
Find the eigenfunctions and eigenvalues of the isotropic bidimensional harmonic oscillator in polar coordinates.
Homework Equations
$$H=-\frac{\hbar}{2m}(\frac{\partial^2}{\partial r^2}+\frac{1}{r}\frac{\partial}{\partial r}+\frac{1}{r^2}\frac{\partial^2}{\partial...
1. I have been trying to plot wavefunctions of QHO for different states with potential energy function using excel. I followed Griffith's Quantum Mechanics, 2nd edition. I got the nature but they have same reference level. Basically I tried to draw fig2.7a (the first one) and got like the second...
Homework Statement
At ##t = 0##, a particle of mass m in the harmonic oscillator potential, ##V(x) = \frac1 2 mw^2x^2## has the wave function:$$\psi(x,0)=A(1-2\sqrt\frac{mw} {\hbar} x)^2e^{\frac{-mw}{2\hbar}x^2}$$
where A is a constant
If we make a measurement of the energy, what possible...
Homework Statement
Hi everybody! In my quantum mechanics introductory course we were given an exercise about the 3D quantum harmonic oscillator. We are supposed to write the state ##l=2##, ##m=2## with energy ##E=\frac{7}{2}\hbar \omega## as a linear combination of Cartesian states...
Homework Statement
Substitute \psi = Ne^{-ax^2} into the position-space energy eigenvalue equation and determine the value of the constant a that makes this function an eigenfunction. What is the corresponding energy eigenvalue?
Homework Equations
\frac{-\hbar^2}{2m}...
Homework Statement
Prove that ##\psi_n## in Eq. 2.85 is properly normalized by substituting generating functions in place of the Hermite polynomials that appear in the normalization integral, then equating the resulting Taylor series that you obtain on the two sides of your equation. As a...
Homework Statement
Show the mean position and momentum of a particle in a QHO in the state ψγ to be:
<x> = sqrt(2ħ/mω) Re(γ)
<p> = sqrt (2ħmω) Im(γ)
Homework Equations
##\psi_{\gamma} (x) = Dexp((-\frac{mw(x-<x>)^2}{2\hbar})+\frac{i<p>(x-<x>)}{ħ})##The Attempt at a Solution
I put ψγ into...
How do you find the wave function Φα when given the Hamiltonian, and the equation:
aΦα(x) = αΦα(x)
Where I know the operator
a = 1/21/2((x/(ħ/mω)1/2) + i(p/(mħω)1/2))
And the Hamiltonian,
(p2/2m) + (mω2x2)/2
And α is a complex parameter.
I obviously don't want someone to do this question...
Homework Statement
How to calculate the matrix elements of the quantum harmonic oscillator Hamiltonian with perturbation to potential of -2cos(\pi x)
The attempt at a solution
H=H_o +H' so H=\frac{p^2}{2m}+\frac{1}{2} m \omega x^2-2cos(\pi x)
I know how to find the matrix of the normal...
Homework Statement
In ##1+1##-dimensional spacetime, two objects, each with charge ##Q##, are fixed and separated by a distance ##d##.
(a) A light object of mass ##m## and charge ##-q## is attached to one of the massive objects via a spring of spring constant ##k##. Quantise the motion of the...
Derivation of energy levels in a quantum harmonic oscillator, ##E=(n+1/2) \hbar\omega##, is long, but the result is very short. At least in comparision with infinite quantum box, this result is simple. I suspect that it can be derived avoiding Hermite polynomials, eigenvalues, etc. I understand...
Homework Statement
Consider a particle with mass m oscillates in a simple harmonic potential with frequency ω. The position, x, and momentum operator, p, of the particle can be expressed in terms of the annihilation and creation operator (a and a† respectively):
x = (ħ/2mω)^0.5 * (a† + a)
p =...
Homework Statement
Part d) of the question below.
Homework Equations
We are told NOT to use the ladder technique to find the position operator as that's not covered until our Advanced Quantum Mechanics module next year (I don't even know this technique anyway). I emailed my tutor and he...
Homework Statement
For a 1-dimensional simple harmonic oscillator, the Hamiltonian operator is of the form:
H = -ħ2/2m ∂xx + 1/2 mω2x2
and
Hψn = Enψn = (n+1/2)ħωψn
where ψn is the wave function of the nth state.
defining a new function fn to be:
fn = xψn + ħ/mω ∂xψn
show that fn is a...
For a 1D QHO we are given have function for ##t=0## and we are asked for expectation and variance of P at some time t.
##|\psi>=(1/\sqrt 2)(|n>+|n+1>)## Where n is an integer
So my idea was to use Dirac operators ##\hat a## and ##\hat a^\dagger## and so I get the following solution
##<\hat...
When I work out $$b^+b$$, I get
$$\widehat{b^+} \widehat{b} = \frac{1}{2} (ξ - \frac{d}{dξ})(ξ + \frac{d}{dξ}) = \frac{1}{2} (ξ^2 - \frac{d^2}{dξ^2}) = \frac{mωπx^2}{h} - \frac{h}{4mωπ} \frac{d^2}{dx^2}$$
So base on what I have about, (9) should be
$$(9) = \frac{hω}{2π} (\frac{1}{2}...
I am trying to perform the operation a on a translated Gaussian, ie. the ground state of the simple harmonic oscillator (for which the ground state eigenfunction is e^-((x/xNot)^2). First, I was able to confirm just fine that a acting on phi-ground(x) = 0. But when translating by xNot, so a...
Homework Statement
We have the lagragian L = \frac{m}{2} \dot{x}^2 - \frac{m \omega x^2}{2} + f(t) x(t)
where f(t) = f_0 for 0 \le t \le T 0 otherwise. The only diagram that survives in the s -matrix expansion when calculating <0|S|0> is D = \int dt dt' f(t)f(t') <0|T x(t)x(t')|0>...
So I've read you can get the corresponding wave function of a quantum harmonic oscillator in momentum space from position space by making the substitution ##x \to k## and ##m \omega \to 1/m \omega##.
However in deriving the TISE for momentum space, I seem to be making a mistake. In momentum...
Homework Statement
Using the equations that are defined in the 'relevant equations' box, show that
$$\langle n' | X | n \rangle = \left ( \frac{\hbar}{2m \omega} \right )^{1/2} [ \delta_{n', n+1} (n+1)^{1/2} + \delta_{n',n-1}n^{1/2}]$$
Homework Equations
$$\psi_n(x) = \left ( \frac{m...
Given the half harmonic potential:
\begin{equation}V=\begin{cases}1/2\omega^2mx^2 & x > 0\\\infty & x < 0\end{cases}\end{equation}What will be the Hamiltonian of the half oscillator?I understand that for x>0 the Hamiltonian will be...
I seem to have two approaches that I've seen and understand, but I can't quite see how they relate.
1. Write a general time evolving state as a superposition of stationary states multiplied by their exp(-iEt/h) factors, and calculate <x>. We find that <x>=Acos(wt+b) as in classical physics (in...
Homework Statement
I must find the average number of energy levels of quantum harmonic oscillator at temperature T, and the answer is given as
I must use Boltzmann distribution and the sum of geometric progression. For finding the average value I must use the equation
<F>=trace(F*rho)...
My problem is described in the animation that I posted on Youtube:
For the sake of convenience I am copying here the text that follows the animation:
I have made this animation in order to present my little puzzle with the quantum harmonic oscillator. Think about a classical oscillator, a...
Homework Statement
There is a harmonic oscillator with charge q and sudenly we turn on external electric field E, which direction is the same as oscillator's. We need to find probability, that particles energy calculated in electric field will be in m state.
n=1, m=2
2. Homework Equations The...
Dear kind helpers,
actually I am not 100% sure whether this is the right place to post, as it is not a homework in the sense of an exercise sheet. But I think it could be because it feels pretty basic and that I should be able to solve it. Though I really searched for a solution but could not...
Definition/Summary
This is the quantum-mechanical version of the classical harmonic oscillator. Like the classical one, the quantum harmonic oscillator appears in several places, and it also appears in the quantization of fields. This article will discuss the one-dimensional version, but it...
The energy changes correspond to infrared, h_bar * w.
Which particles are actually oscillating? The neutrons or the electrons?
Is it the electrons that fill up the stationary states, electronic configuration, or is it the nucleons that fill up the states?
Hi folks!
Apparently
\Psi(x) = Ax^ne^{-m \omega x^2 / 2 \hbar}
is an approximate solution to the harmonic oscillator in one dimension
-\frac{\hbar ^2}{2m} \frac{d^2\psi}{dx^2} + \frac{1}{2}m \omega ^2 x^2 \psi = E \psi
for sufficiently large values of |x|. I thought this...
Homework Statement
For the n = 1 harmonic oscillator wave function, find the probability p that, in an experiment which measures position, the particle will be found within a distance d = (mk)-1/4√ħ/2 of the origin. (Hint: Assume that the value of the integral α = ∫0^1/2 x^2e^(-x2/2) dx is...
A harmonic oscillator with frequency ω is in its ground state when the stiffness of the spring is instantaneously reduced by a factor f2<1, so its natural frequency becomes f2ω. What is the probability that the oscillator is subsequently found to have energy 1.5(hbar)f2ω? Thanks
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 =...