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### Intuition behind creation and annihilation operators? [duplicate]

Here I am talking about Harmonic Oscillators with Hamiltonian $$H=\frac{1}{2m}(p^2+(m\omega x)^2),$$ with eigenstates $|1\rangle,|2\rangle,\ldots$ Many textbooks define the annihilation operator to ...
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### If there are eigenstates of $L_z$ in a degenerate subspace, are there also eigenstates of $L^2$?

The question arises from an exercise but tackles deeper understanding of angular momentum operators. Suppose we have a 2D harmonic oscillator and an infinite square well in the third dimension: \...
112 views

### 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 ...
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### Conceptual understanding of the Quantum Harmonic oscillator

First: When we consider a quantum particle in a harmonic (quadratic) potential we say that this particle is a harmonic oscillator, because it behaves like one. Is this correct? Now let us assume our ...
304 views

### Hamiltonian of quantum harmonic oscillator with $\psi(x)=\delta(x)$: comparison to classical mechanics

I was just reading the question Why can't $\psi(x)=\delta(x)$ in the case of a harmonic oscillator? The accepted answer says that $\psi(x)=\delta(x)$ is a mathematically valid state, though it's not ...
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### 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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### 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 do the ladder operators in harmonic oscillators work?

The Hamiltonian can be diagonalized by transforming $x$ and $p$ to $a$ and $a^\dagger$. I understand how one proceeds from there to find the spectrum of $a^\dagger a$, the ground state $|0\rangle$ and ...
397 views

### Annihilation operator in harmonic oscillator

In Wikipedia's QHO page there is a moment when the following is stated: I don't know why "the ground state in the position representation is determined by $a|0\rangle=0$". I would say that the ...
64 views

### Proportionality of states in quantum harmonic oscillator

What is the justification for $a_{\pm} \psi_{n}$ being proportional to $\psi_{n\pm1}$ in a quantum harmonic oscillator? Here $a_{\pm}$ is the raising/lowering ladder operator.
362 views

### State time evolution of a quantum harmonic oscillator with a Dirac-Delta function as an initial state [closed]

I have a question just like this Phys.SE problem here with a difference that our system is a harmonic oscillator (rather than a free particle). A particle with mass $m$ is connected to a string with ...
508 views

### Eigenstates of sum of creation and annihilation operators

Does the operator $a+a^\dagger$ have eigenstates? If yes, what are they?
Suppose $\mathcal{B}=\{|0\rangle, |1\rangle, |2\rangle, ... \}$ is the energy eigen-basis of a quantum simple harmonic oscillator. I want to create the state \begin{equation} |\Psi\rangle = c_{0}|0\...
### Using creation and annihilation operators to prove the expression for the $n$th excited state in terms of the vacuum state
How does one prove that the $n^{th}$ excited state of a quantum harmonic oscillator (QHO) can be obtained by applying the creation operator $a^{\dagger}$ $n$-times to the vacuum state $\vert 0\rangle$?...