# Questions tagged [harmonic-oscillator]

The term "harmonic oscillator" is used to describe any system with a "linear" restoring force that tends to return the system to an equilibrium state. There is both a classical harmonic oscillator and a quantum harmonic oscillator. Both are used to as toy problems that describe many physical systems.

1,385 questions
Filter by
Sorted by
Tagged with
45 views

### Does the number operator $N$ of the Quantum Harmonic Oscillator commute with $x$? [on hold]

Does the number operator $N$ of the Quantum Harmonic Oscillator commute with $x$?
31 views

### Can anybody tell me the difference between these two velocity?

In Waves i earlier studied that velocity of the wave is given $wa\sqrt{1-y^2}$ w= angular velocity a=amplitude of the wave y=post. of wave at any time and now when i am studying wave optics ...
21 views

### Velocity in SHM

By book defines SHM as Simple harmonic motion is defined as the projection on any diameter of a graph point moving in a circle with uniform speed. but in the next line it says - Moving back ...
6 views

### Frequency of oscillation in Simple U-Tube Manometer by Energy Method

Force Balance Method By balancing accelerating and restoring Force, it's easy to find the angular frequency $\omega$ Restoring Force= (Weight of the extra height of liquid column)=$$\rho*Area*2x*g$$ ...
14 views

### Relation between time period and amplitude of a pendulum [on hold]

What is the relation between time period and amplitude of a simple pendulum? Does time period increase with the increase in amplitude? What will be the graph of this relation?
86 views

### Brownian Harmonic Oscillator

I am trying to solve the below problem, but I am unsure about my attempted solution. Problem statement This problem comes from exercise 2 of the notes on Green's function. My attempted solution ...
11 views

41 views

### Energy eigenfunctions of a truncated harmonic oscillator-like potential

Assume a potential of the form \begin{eqnarray}V(x) &=& \frac{1}{2}m\omega^2x^2,~-x_0\leq x\leq x_0,\\&=& 0,\hspace{2.5cm}{\rm otherwise} \end{eqnarray} where $x_0$ is a finite ...
97 views

### Is there any proof that $F=-kx$?

How do you proof that $F = - kx$? And why is there (-) on the formula(?)
28 views

### Superposed simple harmonic oscillators

When deriving the equation for the superposed amplitude: $$A^2=A_1^2+A_2^2+2A_1A_2 \cos(\phi_2-\phi_1)$$ From $$x_1(t)=A_1 \cos(\omega t+\phi_1)$$and $$x_2(t)=A_2 \cos(\omega t+\phi_2)$$ How do you ...
49 views

38 views

### Complex number representation of a wave

There are some aspects to waves I am confused, for instance in Chapter 11. Fraunhofer Diffraction. The incoming electric fields can be partially expressed as $e^{i(kr-\omega t)}$. I have two ...
39 views

### Pendulum: Time period for effective length varying with angle

For a pendulum whose effective length changes with angle, my approach would be to write the length as a function of the angle: $\ell(\theta)$, then using the torque derivation of the time period I do ...
32 views

### Spring constant of harmonic oscillator [closed]

I got a task from my lecturer to solve a differential equation for a simple harmonic oscillator: $$m{d^2\vec{r} \over dt^2}=-k^2\vec{r}.$$ So far, I have managed to find this equation only in one book....
53 views

### Confusion on Quantum Harmonic Oscillator Eigenvalues

In standard PDE theory, one generates eigenvalues to Sturm-Liouville problems over a finite domain. So, for a wave equation, we have an infinite number of eigenvalues $\lambda_n$ for a Dirichlet ...
26 views

### Elongation of a simple pendulum

One of the questions on this weeks question sheet asks for the maximum elongation of a simple pendulum. The pendulum is set in motion on the moon with f = 0.5Hz. What is meant by the elongation of the ...
24 views

### Hermitian phase operator and quantum harmonic oscillator

I need to apply a hermitian phase operator $\dfrac{1}{\sqrt{1+\hat{a}^\dagger\hat{a}}}\hat{a}$ to the nth harmonic oscillator state, but I have no idea how to interpret the square root of a ...
60 views

### Average values of $\langle n|x_{op}|n\rangle$ and $\langle n|p_{op}|n\rangle$ [closed]

Let an harmonic oscillator described by the hamiltonian $H=p^2/2m+(1/2)mw^2x^2$. I have determined that the average values of the observables $x$ and $p$ in energy eigenstates , $\langle n|x|n\rangle$ ...
56 views

### Energy difference in the Hamiltonian $H_{b}=\hbar\omega(a^{\dagger}a+\frac{1}{2})+\hbar\omega (b-1/2)(a^{\dagger}a^{\dagger} +aa)$ [closed]

Given that a Hamiltonian is on the form $$H_{b}=\hbar\omega(a^{\dagger}a+\frac{1}{2})+\hbar\omega (b-\frac{1}{2})(a^{\dagger}a^{\dagger} +aa)$$ where $b$ is a dimensionless real number in the ...
91 views

42 views

### Finding general equation for motion of a radioactive particle performing SHM [closed]

Let us assume we have a particle of initial mass $m_{0}$ such that a general time $t$: $$m(t) = m_{0} e^{- \lambda t}$$ Now, let us say this particle is attached to a spring of spring constant $k$,...
52 views

### Is it possible to construct a state for harmonic oscillator given the mean energy?

The harmonic oscillator is defined by the mean value energy $\langle E\rangle=\frac{2}{3} \hbar\omega$. Can we have a wavefunction which describes such a state? Any help is appreciated. Is it ...
100 views

### Harmonic oscillator with potential shifted by a constant

I've been thinking a lot about changes to the harmonic oscillator potential, and I was looking into the problem where $$V(x) = \frac{1}{2}m\omega ^2 x^2 + C$$ where $C$ is some positive real ...
39 views

### Does changing the angle of a pendulum also shift the coordinate plane w.r.t which we give rectangular components to the $mg$ vector?

So given a simple pendulum, which makes an angle of 0 with the vertical axis in it's resting position.Now the pendulum is moved to a side by an angle $\theta$ with the vertical axis. The components of ...
46 views

### Does resonance just depends upon the frequency of the external periodic force and the natural frequency of an object?

I am a little confused about the phenomenon of resonance, I read that it occurs when the frequency of an external force matches the natural frequency of an object. So, it was given that soldiers ...
54 views

### $Q$ factor of a pendulum

according to the definition of the Q-factor of damping, it is given by: $Q = 2\pi\frac{Energy \; Stored}{ Energy \;Dissipated \; per \;cycle }$ Q = 1⁄2 --> Critical damping Q > ​1⁄2 --> Over ...
39 views

### Canonical quantisation harmonic oscillator

I have a question on the canonical quantisation as described at the linked wiki page: https://en.wikipedia.org/wiki/Quantum_field_theory#Canonical_quantisation we take the displacement of a classical ...
36 views

### Why is resultant displacement in an composition of simple harmonic motion the sum of individual displacements?

I recently came across the concept of the composition in simple harmonic motion. A paragraph says that: If $$x_1 = A_1sin(\omega t)$$ $$x_2 = A_1sin(\omega t + \phi)$$ Then, the resultant ...
52 views

### Why is the force of gravity positive for an oscillating spring?

When analyzing the movement of a weight attached to a spring, many sources set up the force equation using newton’s second law as follows. $$mg-k(L+x)=ma$$ where $L$ is the length that the mass $m$ ...
64 views

### Quantum harmonic oscillator hamiltonian in terms of the parity operator

Can you write the quantum harmonic oscillator hamiltonian $$H = -\dfrac{\hbar^2}{2m}\dfrac{d^2}{dx^2}+\dfrac{1}{2}m\omega^2x^2$$ in terms of the parity operator $P$?
If, for a (not necessarily simple harmonic) oscillator I have that $$\frac{dx}{dt} = G(x)$$ then I can express the period of motion as $$\int_{0}^{T/4} dt = \int_{0}^{X_{max}} \frac{dx}{G(x)}.$$ What ...
I am trying to find the period of small oscillations of the potential $$V(x) = \frac{1}{2}m\omega_0^2(x^2-bx^4)$$ It is given that the particle oscillates between $-a$ and $a$ for some \$a < \...