All Questions
Tagged with harmonic-oscillator hamiltonian
59 questions
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Physical interpretation and validity of QHO hamiltonian term: $\hat{a}\hat{n} + \hat{n}\hat{a}^{\dagger}$ [closed]
For a quantum harmonic oscillator can we have a Hamiltonian term of the following form, and what would be its physical interpretation:
$$\hat{a}\hat{n} + \hat{n}\hat{a}^{\dagger}$$
10
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3
answers
1k
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Quantum harmonic oscillator meaning
Imagine we want to solve the equations
$$
i \hbar \frac{\partial}{\partial t} \left| \Psi \right> = \hat{H}\left| \Psi \right>
$$
where $$\hat{H} = -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial ...
- 221
0
votes
1
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36
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The time-derivative of the Hamiltonian for a 1D harmonic potential [closed]
I do not understand how to take the time derivative of the following Hamiltonian $\hat{H}(t) = \frac{\hat{p}^2}{2m} + \frac{1}{2}m\omega^2(\hat{x}-a(t))^2$, where $a(t) = v_0t$. For instance how does ...
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1
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234
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Motivation for introducing ladder operators for the simple harmonic oscillator in quantum mechanics
I am teaching a quantum mechanics course and I have to explain the simple harmonic oscillator. I am familiar with the introduction of ladder operators and the consecutive proofs that show that we can ...
- 4,737
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3
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280
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It seems that expectation value of $H$ on coherent states is independent of time? But why?
Let's say the particle is in the state $| \psi(0) \rangle = \exp(-i\alpha p/\hbar) |0 \rangle$, where $p$ is the momentum operator.
I have to show that $| \psi(0) \rangle$ is a coherent state and to ...
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0
answers
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Quantum Harmonic Oscillator: find a constant $\beta$ such that $U=\exp(\beta a^{\dagger} - \beta^*a)$ diagonalize $H$ [closed]
Given Hamiltonian of Quantum Harmonic Oscillator,
$$H = \frac{p^2}{2m}+\frac{1}{2}m\omega^2 x^2-\gamma x$$
I have to find a constant $\beta$ such that the unitary operator $U=\exp(\beta a^{\dagger} - \...
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1
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What is the probability to find the system in the ground state? [closed]
I previously posted a question related to this Hamiltonian, but the original concern was different:
We examine the following Hamiltonian:
\begin{equation}
H = \frac{p^2}{2m} + \frac{1}{2}m\omega^2x^2 -...
user372470
3
votes
2
answers
372
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Does the state change, when the Hamiltonian changes?
Consider the Hamiltonian
\begin{equation}
H = \frac{p^{2}}{2m} + \frac{1}{2} m\omega^{2}x^{2} - \theta(t) qEx
\end{equation}
where $\theta(t)$ is $0$ for $t = 0$ and $1$ for $t > 0$. If at $t = ...
user372470
0
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2
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145
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Quantum harmonic oscillator as the potential becomes zero
I'm having a question concerning the quantum harmonic oscillator: If, for instance, $\omega\to 0$ the Hamiltonian
$$
\hat{H} = \frac{\hat{p}^2}{2m} + \frac{1}{2}m\omega^2 x^2\tag{1}
$$
becomes that of ...
- 19
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1
answer
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How to diagonalize a single particle hamiltonian? [closed]
$$H=\hbar\omega \left(a^\dagger a+\frac{1}{2}\right)+\hbar \omega_0\left(a^\dagger+a\right)$$ How to diagonalize $H$ and find its eigenenergies?
- 11
4
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1
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Interpretation of this Hamiltonian
I'm studying a system with the following Hamiltonian $$ H = \frac{1}{2}P^TAP + \frac{1}{2}Q^T B Q$$ where $P,Q$ are canonical variables (4-vectors) and $A,B$ matrices such that $A = A^\dagger$ and $B =...
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3
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What is the form of an electrical oscillator Hamiltonian?
I found this https://en.wikipedia.org/wiki/Harmonic_oscillator in my search results when I search for "electrical oscillator Hamiltonian" and some other things too. But none of them answer ...
- 2,042
1
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1
answer
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Eigenstates for quantized oscillator [closed]
Hi I am new to solid state physics and am reviewing a prior knowledge section and would like some clarification. The following appeared in the course notes:
From my understanding, Eigenstates are ...
1
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0
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Using variation principle on quantum oscillator with general potential
Consider a general bounding potential $V(x)$. The hamiltonian is $$H = \frac{p^2}{2m} + V(x).$$ We want to apply the variation principle in equation $$F\leq F_0+\langle H-H_0\rangle_0.$$ $\langle\...
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1
answer
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Why the first-order derivative is missing when composing a Hamiltonian of simple harmonic oscillator by the lowering and the raising operators? [closed]
Given the lowering operator ($a$) and the raising operator ($a^\dagger$)
$$\begin{align*}
a &= \frac{1}{\sqrt{2m \hbar \omega}}\left(-i \hbar \frac{\partial}{\partial x} - i m \omega x\right) \\
a^...
- 277
2
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0
answers
40
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Propagator for radial force field?
The propagator $K(x,y;t)$ is well known for the (1D) harmonic oscillator:
$$H = -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} + \frac{m}{2}\omega^2 x^2$$
is there a simple closed form solution ...
user84158
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0
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Canonical transformation of the harmonic oscillator‘s Hamiltonian [closed]
I could deduce the Hamiltonian of the damped harmonic oscillator:
$$
H=\frac{p^2}{2m}e^{-2 \gamma t}+\frac{m \omega_0^2 q^2}{2}e^{2 \gamma t}
$$
Using the canonical transformation $Q=e^{\gamma t}q, P=...
- 522
3
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1
answer
324
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Modified quantum harmonic oscillator spectrum and eigenstates
I am trying to find the eigenstates/eigenvalues of the following Hamiltonian
$$
\hat{H} = \hbar \omega \Big(\hat{a}^{\dagger}\hat{a}+\frac{1}{2}\Big)+A\big(\hat{a}^{\dagger}\hat{a}^{\dagger}+\hat{a}\...
- 745
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I'm getting the wrong Hamiltonian of the quantum oscillator
First, I generalised the oscillator's Hamilton's equations to complex variables:
$$\frac{dz_1}{dt}=\frac{\partial (z_1^2+z_2^2)}{\partial z_1}=2z_1$$
$$\frac{dz_2}{dt}=2z_2$$
So the real world ...
- 959
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1
answer
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Eigenstates harmonic oscillator with mass matrix
Consider the 2D harmonic oscillator
$H = \langle \nabla, M\nabla \rangle+ \vert x \vert^2$ where $x \in \mathbb R^2$ and $M$ is a symmetric mass matrix with strictly positive eigenvalues.
Is it known ...
- 101
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2
answers
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Time Derivative of the Hamiltonian for a Quantum Simple Harmonic Oscillator
I am reading an article on quantum refrigerator. Here is the link of the article. The arXiv version is available here. The working medium is an ensemble of non-interacting particles in a harmonic ...
- 439
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2
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Hamiltonian of two coupled oscillators
Lets say I have this system:
Two different masses with three different springs.
It's not very nice to do, but I can find the eigenvalues of this system (It's not nice because the two masses are ...
- 11
1
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2
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616
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Confusion in Quantum Harmonic Oscillator
I am confused with the meaning of the particle number of a quantum harmonic oscillator. Classically, the Hamiltonian of harmonic oscillator in phase space is defined as follows:
$$H = \frac{p^{2}}{2m} ...
- 435
1
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3
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Why is $\langle n| (\hat{a}+\hat{a}^{\dagger})^2|n\rangle=2n+1$ for the QM harmonic oscillator? [closed]
Consider a one-dimensional quantum-mechanical simple harmonic oscillator of mass $m$ and potential energy $\frac{kx^2}{2}$. The energy levels of this system are $E_n=(n+\frac{1}{2})\hbar\omega $ for $...
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1
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Statistical weight for $N$ harmonic oscillators in microcanonical ensemble
I would like to compute the statistical weight for the microcanonical ensemble for $N$ harmonic oscillators.
To do that i use the hamiltonian of the harmonic oszillator:
$$H(q,p)=\sum\limits_{i=1}^N \...
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0
answers
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Transmon: why do we need unharmonic hamiltonian to isolate energy levels? [duplicate]
With a quantum harmonic oscillator, we cannot isolate energy levels, e.g. to create a qubit. We need to embed anharmonicity, to get unevenly-spaced energy levels and so making them distinguishable.
...
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5
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How can zero-point energy have any measurable effect when it is just a constant offset to the Hamiltonian?
In classical (say Hamiltonian) mechanics, adding a constant energy offset has no effect on the dynamics of the physical system. One way to understand this might be to understand that Hamilton's ...
- 15.2k
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Peskin and Schroeder, where is the mass in the denominator of the simple harmonic oscillator Hamiltonian?
This relates to page 20 of Peskin and Schroeder.
They state that the Fourier transform of the Klein-Gordon field satisfies the following:
$$\left[\frac{\partial^2}{\partial t^2}+(|\vec p|^2+m^2)\right]...
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2
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Deriving the Hamiltonian for a simple pendulum using mechanical momentum as a free parameter
So when we covered the derivation of a simple pendulum we , and from what ive found on the web, defined our free parameter as $q=L\theta$ and arrive at the Hamiltonian for a Harmonic oscillator.
But ...
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1
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How to determine the ground state of quantum harmonic oscillator like Hamiltonian?
For the time-dependent Hamiltonian
$$H = \frac{\hat{P}^2}{2m} + \frac{1}{2} m\omega^2\hat{X}^2 + m\omega^2vt\hat{X} +v\hat{P}$$
I would like to calculate the ground state, more precise, the stationary ...
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1
answer
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Interpreting Hamiltonian of single-mode squeezing
Hamiltonian represents energy. I can understand this when considering about harmonic oscillator, whose Hamiltonian is expressed as:
$$ \hat{H} = \frac{1}{2m}\hat{p}^2 + \frac{m\omega^2}{2}\hat{q}^2$$
...
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2
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Is symmetrization $xp-px$ required for commutation $[H,x]=0$?
Given a Quantum Hamiltonian: $$\hat{H}=ax^2+bp^2$$ It does not commute with either $x$ or $p$. Suppose we have a Hamiltonian :$$H = k \hat{p}\hat{x}$$ why do we need it to be: $$H = k (\hat{p}\hat{x} -...
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0
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In context of defining a hamiltonian of a system, can someone please explain the meaning of zero-point oscillations/vibrations?
I came across this term while studying electron delocalization in amorphous semiconductors. The delocalization of the electronics wavefunction is explained in terms of zero-point oscillations.
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1
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408
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Calculating exact energy levels of perturbed Hamiltonian
I wish to find the exact energy levels of the following perturbed hamiltonian.
$$\hat{H}=\frac{p^2}{2m}+\frac{m\omega^2}{2}x^2+\alpha x+\beta p^2.$$
I believe that it can be solved by using the ...
- 558
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0
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401
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Solution to two non-coupled quantum harmonic oscillators
Given the following Hamiltonian:
$$\hat H = -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x_1^2} -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x_2^2} + \frac{1}{2}m\omega_1^2x_1^2 + \frac{1}{2}m\...
- 11
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1
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Why do we consider only one mass when solving linear harmonic oscillators in quantum physics?
While solving the Hamiltonian, books concentrate on the horizontal flow with only one mass attached to the string. Isn't there any consequences if we add more masses and why is friction always ignored?...
-2
votes
2
answers
435
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Is a quantum harmonic oscillator always infinite dimensional?
Let us assume we have a quantum particle in a harmonic potential with the Hamiltonian
$$H = \sum_n n \omega |n\rangle\langle n|$$
If I am not mistaken.
Now when talking about harmonic oscillators ...
- 415
1
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4
answers
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Why we neglect the $\hbar ω/2$ in the Hamiltonian of the the Electromagnetic Field?
After the quantization of the electric and the magnetic field, we get the Hamiltonian of the electromagnetic field:
$$H= \hbar ω(a^{\dagger}a +1/2) .$$
with $\hbar$ the planck constant and $a^{\...
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1
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Harmonic oscillator energy difference between $(n+\frac{1}{2})h \omega$ and $(n+\frac{1}{2})\hbar \omega$
When I was studying the Harmonic Oscillator using the Schrödinger equation, I was told in lectures to pay attention to the units.
There were 2 different equations given for the Energy of a Harmonic ...
- 295
4
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3
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4k
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What is the Hamiltonian in the "energy basis" for a simple harmonic oscillator?
My textbook says that for a simple harmonic oscillator the Hamiltonian can be expressed in the "energy basis" in this way:
$$\hat H=\hbar\omega\bigg(\hat a^{\dagger}\hat a + {1\over 2}\bigg).$$
I ...
- 795
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1
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345
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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
$$
\hat{H} = \left(\hat{a}^{\dagger}\...
- 41
1
vote
0
answers
360
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One Hamilton Operator for two independent harmonic oscillators
If we consider two independent harmonic oscillators (identical too a two dimensional harmonic oscillator), the hamilton operator is
$$
H = \frac{p_1^2}{2m_1}+\frac{1}{2}m_1\omega_1^2x_1^2
+ \frac{...
- 11
2
votes
1
answer
2k
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Systematic way of decoupling a coupled oscillators Hamiltonian
I have been faced with the Hamiltonian $$H = P_1^2/2m_1 + P_2^2/2m_2 + (k/2) x_1^2 + (k/2)x_2^2 + (K/2)(x_1-x_2)^2$$
I'm trying to find a systematic way to decouple it other than guessing. So, I wrote ...
- 23
2
votes
1
answer
2k
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Transforming simple Hamiltonian to interaction picture
I am trying to follow the math in a paper, and in it they do a lot of transforming into the interaction frame. It has been awhile since I have done these kind of calculations explicitly by hand and I ...
- 1,001
-1
votes
2
answers
1k
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Hamiltonian approximation of the Coulomb interaction energy of two charged oscillators
I'm adding an excerpt from the book Introduction to Solid State Physics 7th edition by Charles Kittel.
I don't see how they arrived at the approximation of the Hamiltonian (2) by expanding it. If $...
- 443
1
vote
2
answers
1k
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Hamiltonian of Klein-Gordon Field
The Hamiltonian of the Klein-Gordon Field may be written $$H=\int\frac{d^{3}p}{(2\pi)^{3}}\frac{1}{2\omega_{\mathbf{p}}}\omega_{\mathbf{p}}\left(a^{\dagger}(p)a(p)+\frac{1}{2}(2\pi)^{3}2\omega_{\...
- 411
2
votes
4
answers
3k
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Recovering the Hamiltonian from ladder operators
The Hamiltonian for the quantum harmonic oscillator is
$$\hat{H}=-\dfrac{\hbar^2}{2m}\dfrac{\partial^2}{\partial x^2}+\dfrac{1}{2}m\omega^2 x^2$$
and one can try to factorise it by writing down what ...
- 1,389
1
vote
2
answers
473
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Harmonic oscillator hamiltonian (QFT)
I have a little doubt about the harmonic oscillator hamiltonian written at the beginning of Peskin & Schroeder's "An introduction to quantum field theory"; I enclose the picture of the page.
...
- 649
-2
votes
1
answer
241
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Canonical Quantization of harmonic oscillator
I have a system of two particles with the usual Lagrangian,
$$L=\frac12M_1{\dot{x_1}}^2+\frac12M_2{\dot{x_2}}^2-\frac12k({x_1}^2+{x_2}^2)$$
I want to find the quantum Hamiltonian of the system. I ...
3
votes
1
answer
441
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What is meant by taking the partial derivative of the Hamiltonian in this situation?
I'm doing a computation involving the quantum mechanical harmonic oscillator, and I have an expression of the form $\frac{\partial}{\partial \omega} \hat{H}$ where
$$\hat{H} = \frac{1}{2m} \left( - \...
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