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The term "harmonic oscillator" is used to describe any system with a "linear" restoring force that tends to return the system to a 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.

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A harmoinic oscillator in the Heisenberg picture

Considering the Hamiltonian of a harmonic oscillator \begin{equation} H=\frac{p^2}{2m}+\frac{m\omega^2 x^2}{2}, \end{equation} the time evolution of the Heisenberg picture position and momentum ...
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Energy level in harmonic oscillator [duplicate]

ground state is always non degenrate in harmonic oscillator in quantum state, why?
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Is there a proof for why the acceleration of an object undergoing simple harmonic motion related to angular velocity squared?

Many textbook says the defining equation of the acceleration of an object undergoing simple harmonic motion is $$a= -\omega^2 \times x.$$ Is there a reason as to why acceleration is related to $\...
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Number operator and its eigenvectors [duplicate]

So the eigenvalue problem for a Number operator defined in terms of creation and annihilation operators ( $N = a^{\dagger}a$ ) is $$N|n>n|n>$$ If we act with $N$ on $a|n>$ and use ...
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1answer
51 views

A particle is in a harmonic oscillator potential. Which are the possible spreads for a simultaneous measurement of the momentum? [on hold]

A particle is in a harmonic oscillator potential, not in the ground state. The position of the particle is known with an rms spread of $1\mathring{\text{A}}$. Which of the following are possible ...
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24 views

understanding lock-in measurements (XY display mode)

I am trying to understand the lock in measurement performed here. Mainly, I am trying to understand how they were able to choose the phase offset based on the plot (first image below). In the ...
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1answer
32 views

What is the effect of mass on resonance amplitude?

When a system is undergoing forced oscillations, why does reducing the mass of the system cause the frequency response curve to shift downwards? I encountered this problem in a practice paper, but I ...
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1answer
52 views

Using a single harmonic oscillator to implement a quantum gate. Confusion over concept

I'm trying to simulate a quantum gate operation in mathematica using a harmonic oscillator and I have some confusion with how the physical system relates to the theory. This may be a bit long winded ...
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1answer
62 views

Can anyone explain the harmonic oscillator (in context to quantum mechanics) 2.3 (Griffiths) using Taylor series?

At the end he concludes $V(x) = V''(x_0)(x-x_0)^2$. How does he get to know that the rest are $0$? How does he conclude $V''(x_0) = k$. Please try to explain in easy ways and tough vocabulary. I don't ...
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40 views

Damped harmonic oscillator

The book I am using to study classical mechanics refers to "characteristic time" of an over damped system. I don't quite understand what they mean by this. I know relaxation time of a underdamped ...
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54 views

Modelling a pendulum with physical restrictions on it's range of motion

I'm currently working on a project based on suspension bridges and their oscillations. I've got an equation of motion for the movement of a pendulum as shown in the first image, I then wanted to be ...
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3answers
38 views

Gravity’s effect on a vertical spring-block simple harmonic oscillator

I just found a question in my textbook which asked how the period of the vertical oscillation will change if the spring and block system is moved to the moon, and the gravity due to acceleration is ...
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107 views

meaning of “sufficiently small” in approximations to behaviour of a Harmonic Oscillator

So in my classical mechanics book it states: "For any sufficiently small displacement, any system of this kind behaves like a harmonic oscillator." When discussing SHO. So I am curious what is ...
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1answer
69 views

Why do materials obey Hooke's law? [duplicate]

Why do materials extend proportionally to the force exerted on them (Hooke's law)? I thought that when materials are compressed or extended under force, their atoms become closer or further apart; ...
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41 views

A deformed Creation and Annihilation Operetors

When we talk about para-bose oscillators under a chemical potential $\mu$ we might bring up the Eq. $$\frac{1}{e^{\beta(E-\mu)}-1}$$ Seems natural to think that now the vacuum energy of the ...
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31 views

How to turn a 2D wave problem into a 1D oscillator problem?

I would like to turn a 2D problem about a driven wave into a 1D problem about a driven oscillator. Essentially, I would like to be able to turn the wave equation into an equation which describes the ...
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40 views

Justification for using wave equation for describing a phenomena

I have recently started learning about waves. We didn't really formally describe what a wave is, but instead started by looking at a concrete example namely harmonic sinusoidal waves in 1d. We then ...
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1answer
59 views

Would a spring ever stop?

It is not difficult to show from Newton's second law $$m\ddot x = -kx - b\dot x $$ that an underdamped spring has the equation of motion quantified by $$x(t) = c_1e^{-\beta t}\sin\left(\omega t\right) ...
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2answers
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Mass hanging by a spring [closed]

I have a mass hanging from the ceiling by a spring, with costant $k$, and wih gravity $g$. Using analytical mechanics I got to the differential equation: $$ \ddot{x} + \frac{k}{m}x + g = 0 $$ The ...
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Is the time period of a fixed string always a resonant time period?

Consider a string held fixed at $x=0$ and $x=L$. This string has a harmonic series associated with it with each harmonic time period, $T_n$, given by: $$T_n=\frac{2}{n}\int\frac{dx}{v(x)},$$ where $v(...
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1answer
48 views

Thermal wave function of the harmonic oscillator - proving that it's a gaussian?

I'm a bit stumped trying to prove this. I've computed the probability density for a thermal density matrix for the quantum harmonic oscillator, namely $$ \rho(x) = \frac{\sum_n^\infty e^{-\frac{\hbar\...
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2answers
41 views

Determine period and angle of harmonic oscillator with $x = 2 \pi \sin(120 \pi t + 3.2 t)$ [closed]

A particle moving under simple harmonic motion has displacement $$x = 2 \pi \sin(120 \pi t + 3.2 t) \, .$$ How can I determine the period and the phase angle?
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1answer
56 views

Why are oscillations so ubiquitous in nature? [duplicate]

I'm aware that you can always approximate a potential by a quadratic term. But is this the most 'fundamental' reason for the pervasiveness for oscillations?
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65 views

What is the relationship between excitation and resonance?

From Resonance (particle physics) - Wikipedia: In particle physics, a resonance is the peak located around a certain energy found in differential cross sections of scattering experiments. These ...
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2answers
233 views

Do the ladder operators $a$ and $a^\dagger$ form a complete algebra basis?

It is easy to construct any operator (in continuous variables) using the set of operators $$\{|\ell\rangle\langle m |\},$$ where $l$ and $m$ are integers and the operators are represented in the Fock ...
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1answer
44 views

Can the precise form of a quantum field Hamiltonian be determined by harmonic oscillator analogy?

In one of my other questions What is the precise formal correspondance between an oscillator and a quantum field? , I was helpfully given the exact form of the well discussed analogy between the ...
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61 views

Time evolution of state $|\psi (0)\rangle$ for given Hamiltonian

How does one evolve the base state $|\psi (0)\rangle$ to $|\psi (t)\rangle$ with the Hamiltonian $\hat H=\hbar \omega (\hat a ^\dagger \hat a+1/2)$ when $|\psi\rangle (0)=|x_0\rangle$ So $X|x_0\...
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1answer
52 views

Eigenfunctions of Hamiltonian (question about the book “Quantum Field Theory for the Gifted Amateur”)

In the book "Quantum Field Theory for the Gifted Amateur" by Blundell and Lancaster, (page 21) the Hamiltonian (when discussing the number operator) is given by $$ \hat{H} = \left(\hat{a}^{\dagger}\...
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25 views

finding $U(t)\hat a U^\dagger(t) $ when U(t) is the time evolution operator of the Hamiltonian $H=\hbar\omega(\hat N-0.5)$

I need to find $U(t)\hat a U^\dagger(t) $ when U(t) is the time evolution operator of the Hamiltonian $H=\hbar\omega(\hat N-0.5)$?
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3answers
287 views

Simple harmonic oscillator by operators

I'm reading simple harmonic oscillator problem in "Modern Quantum Mechanics" by J.J. Sakurai. The approach is by defining the annihilation ($a^{t}$) and creation ($a$) operators, then a number ...
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1answer
43 views

Dynamical variables in a quantum oscillator

Can someone please explain how we get the first equality (1.118)? (Here $\omega_p$ is the frequency of the quantum harmonic oscillator, whose 'dynamical variable' is $q_p$)
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Why can waves (audio, eletromagnetic, etc.) be represented by a circle?

I found an intuitive reason for the sine function, but want one that can be used for all kinds of waves. I've found a good explanation summarized in the following figure: It is logical because a ...
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What is the mathematical basis behind using a chirp signal to determine the resonant frequency of a second order differential system?

It is a common practise for engineers to try to determine the resonant frequency of a system through a chirp signal. Given a damped oscillating system with displacement $x$, driven by a chirp signal ...
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Expectation values in multiple dimensions

If we consider for example two coupled quantum harmonic oscillators with wavefunction $\psi(x_1,x_2,t)$, to calculate the expectation value of $x_1$ , do I have to integrate from $-\infty$ to $\infty$ ...
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Waves on a slack string

hello im doing an experiment on waves and i need to know how to calculate the wavespeed of a wave in a circular slack string hanging from a pulley. Is there an equation for this? Is there tension in ...
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Similarity between the period of a wave and the period of its source

I have studied only some basic stuff on mechanical waves Consider a string attached to wall at one side and you're holding the other side so that the string is taut. What I've noticed is that when ...
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2answers
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Why do we only solve for even/odd solutions to the quantum harmonic oscillator?

In Griffiths, he writes the following: For normalizable solutions,the power series of h(z), where phi = h(z)e^(-z^2)/2, must terminate. For some highest subscript, call it n, $a_{n+2}=0.$ This ...
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Renormalization of Harmonic Oscillator

In Appendix A, Polchinski does the Euclidean path integral for the Harmonic oscillator. After he Pauli-Villars regularizes the determinant of the kinetic term, he obtains the following expression (A.1....
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Why does Wikipedia state that angular velocity is equal to angular frequency? [duplicate]

Isn't Wikipedia wrong? At the Wikipedia article Angular frequency it says that angular frequency and angular velocity are equal. But how on earth are they equal? Angular velocity is changing all ...
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3answers
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When the oscillator is a system with an angle can we define the angular frequency to be the radians per unit time covered by the system itself?

I read on stackechange that in springs or any one dimensional oscillator the angular frequency is just describing a rate of angle change in the associated circle on which it's projected. Something ...
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162 views

Is angular frequency the same as angular velocity or are they different?

I know there are duplicates. But the answers seem to disagree and also I have more specific questions related to this title. First of all, most questions on this site which ask this question have ...
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Continuation of my previous questions on coupled harmonic oscillator

Coupled many body quantum harmonic oscillator in 3 Dimension $$H=\sum_{j}\frac{p^{2}_{j}}{2m}+\sum_{i<j}\frac{1}{2}k(R_{i}-R_{j})^{2}$$ To solve this problem I have used orthogonal jacobi ...
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1answer
56 views

Coherent States and their existence

In my Quantum Mechanic class, I have learned that to solve for any quantum system, we solve the time independent Schrodinger equation(for time independent Hamiltonian) and then apply the time ...
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1answer
30 views

Exclusion Principle for Two $e^-$ in Harmonic Oscillator Ground State

Suppose we have two $e^-$ in a one Dimensional Harmonic Oscillator with total spin $1$. I am looking for the ground state (and I've read this question Ground State Wavefunction of Two Particles in a ...
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energy per particle in the ground state for non-interacting identical particles moving in external linear harmonic oscillator potential

Consider N non-interacting identical particles moving in an external linear harmonic oscillator potential. I want to calculate the energy per particle in ground state and show that It is constant if ...
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Obtaining the number of quanta for a system of harmonic oscillators

So I need to find the entropy of a system made up of two harmonic oscillators having natural frequency $\omega_0$ and $2\omega_0$. The system is said to have a total energy of $E=(n+\frac12)\hbar\...
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Creating sympathetic resonance in a guitar string with an electromagnet

I have an idea for a potential new type of reverb and I wanted to know if it was possible/practical. The idea is to have an electromagnet in the middle of a guitar string(s) which will hopefully ...
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1answer
39 views

Is there a conservative force acting on a spring that oscillates in SHM?

If I were to stretch a spring, then I am doing positive work in order to increase the potential energy stored in the spring. Since the equation for the potential energy in this case is given by $$U(x)=...
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2answers
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Uniqueness of general solution to SHO

This may be a duplicate, though I have searched and not found this question answered, and it may also belong more on Mathematics Stack exchange than here -- in which case I'll transfer. My question ...
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One Hamilton Operator for two independent harmonic oscillators

If we consider two independent harmonic oscillators (identical too a two dimensional harmonic oscillator), the hamilton operator is $$ H = \frac{p_1^2}{2m_1}+\frac{1}{2}m_1\omega_1^2x_1^2 + \frac{...