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votes
1answer
257 views

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 ...
2
votes
0answers
49 views

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: \...
-2
votes
2answers
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 ...
0
votes
0answers
51 views

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 ...
3
votes
4answers
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 ...
3
votes
1answer
174 views

Why can't $\psi(x) = \delta(x)$ in the case of Harmonic oscillator?

In the analysis of Harmonic Oscillator, it is claimed that $\langle\hat H\rangle$ cannot be zero, why is it so? I mean $\hat H = \frac{ \hat p^2 }{2m } + \frac12 k \hat x^2$, and $$\left<x^2\...
1
vote
4answers
76 views

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^{\...
7
votes
1answer
223 views

Proof $\exp(-\beta H)$ trace-class operator

Let $H=\frac{p^2}{2}+\frac{x^2}{2}\, : D(H) \to L^2(\mathbb{R})$ be the Hamiltonian of the harmonic oscillator with $m=\hbar=\omega=1$. Prove that $\exp(-\beta H)$ is a trace-class operator if $\beta&...
2
votes
2answers
108 views

Relation between destruction/creation operators of harmonic oscillator in QM and Second Quantization

It is well known that in elementary QM the so-called destruction/creation operators $$ a_i = \frac{Q_j + i P_j }{\sqrt{2}}, \quad a_i^* = \frac{Q_j - i P_j }{\sqrt{2}},$$ are introduced when ...
0
votes
2answers
56 views

Proving the raising and lowering of the raising and lowering operator

I am given a written proof of $\hat A^{\dagger}[u_n] = \sqrt{n+1} \ u_{n+1}$, and from it, and told to similarly prove $\hat A[u_n] = \sqrt{n} \ u_{n-1}$. However, in the written proof for $\hat A^{\...
3
votes
3answers
291 views

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 ...
0
votes
1answer
72 views

If $\psi$ acting on $a_+$ and $a_-$ operator just moves up and down the ladder, why is $[a_-, a_+] = 1$ and not 0? [duplicate]

If $\psi$ is acted upon by both the operators one by one, it should return the same wave function. Thus order in which you increase or decrease the energy shouldn't matter. Then why is it so that the ...
8
votes
2answers
394 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 ...
0
votes
1answer
72 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}\...
2
votes
3answers
341 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 ...
0
votes
1answer
57 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 $q_p$)
2
votes
3answers
266 views

How does one calculate the position eigenvalues of the matrix corresponding to the position operator?

The matrix representation corresponding to the position operator is: $$x = \sqrt{\frac{\hbar}{2 m \omega}} \left[ \begin{array}{ccccc} 0 & \sqrt{1} & 0 & 0 & \cdots \\ \sqrt{1} & ...
1
vote
2answers
1k views

How can the expected value $\left<x\right>$ of a given state in an harmonic oscillator differ from $0$?

In a standard harmonic oscillator potential I have the state $\left|\Psi\right> = \frac{1}{\sqrt{2}}(\left|0\right> + \left|1\right>)$ and if I calculate the expected value $\left<x\right&...
5
votes
3answers
254 views

Ladder operator identity for $\langle n | (a+a^\dagger)^k | m \rangle$

I would like to know if there is a convenient identity (and what it is) for $$\langle n | (a+a^\dagger)^k | m \rangle$$ where $| n \rangle, \, | m \rangle$ are energy eigenstates of a simple ...
5
votes
2answers
165 views

Non-integer powers for the quantum harmonic oscillator ladder operators and spectrum uniqueness

Introduction (The idea to this question came from my answer to Uniqueness of quantum ladder for the harmonic oscillator) The Hamiltonian $H$ for quantum harmonic oscillator can be written in terms ...
5
votes
2answers
423 views

Coherent states of Quantum harmonic oscillator

Coherent states of Quantum harmonic oscillator . The Hamiltonian of Quantum harmonic oscillator is $H=(a^+ a+\frac{1}{2})\hbar \omega$,$a=\sqrt{\frac{m \omega}{2 \hbar}}(\hat{x}+\frac{i \hat{p}}{m \...
0
votes
1answer
312 views

Harmonic Oscillator - Zero Point Energy and the Correspondence Principle

I have been studying the harmonic oscillator in quantum mechanics. I fully understand the origin of the zero-point energy and how it can be mathematically shown using the uncertainty principle that ...
0
votes
1answer
175 views

If eigenstate for a Hermitian operator are orthonormal why are the energy eigenstates not so?

My Quantum Mechanics notes says: We found the orthonormality relation holds for any Hermitian operator eigenstates: Upon reading this I would assume that this holds for Energy eigenstates too. Then ...
0
votes
0answers
180 views

Quantum harmonic oscillator: How can we know that the lowest eigenvalue for the operator $A^{\dagger}A$ is zero and not a positive number? [duplicate]

With operator methods we can set the Hamiltonian of the harmonic oscillator in the following form: $$\hat{H}=\hbar \omega(A^{\dagger}A+1/2).$$ My question is that how can we know that the lowest ...
0
votes
2answers
347 views

Truncated harmonic oscillator gives wrong Heisenberg uncertainty?

When I calculate the fundamental commutator, $[x,p]$, in finite-dimensional Hilbert space (Heisenberg picture), the result is not proportional to identity, e.g. if I put $n=2$ (operators in natural ...
1
vote
1answer
425 views

How do these raising and lowering operator combinations give the following values? [closed]

Why do these raising and lowering operator combinations give the following answers? $\langle n^{(0)}\mid\eta x^4\mid n^{(0)}\rangle$ $ = \eta\left(\frac{\hbar}{2m\omega}\right)^2\langle n^{(0)}\mid\...
2
votes
2answers
199 views

How many wave functions can be represented as a superposition in a simple harmonic oscillator?

I'm teaching myself about QM, but there are something really puzzling me about the simple harmonic oscillator: $$H=\frac{p^2}{2m}+\frac{m\omega^2x^2}{2}.$$ I've learned how to use ladder operators to ...
-3
votes
1answer
128 views

How do you describe a quantum state vector mathematically?

Multiple sources I have used have said that a quantum state vector $|x\rangle$ can be taken to be or approximated to be: $$|ψ\rangle = \mathrm e^{-\omega x^2/2}$$ Where $\omega$ is the angular ...
0
votes
1answer
253 views

Normalising the Quantum Harmonic Oscillator

I have been working on the quantum harmonic oscillator with ladder operators and I am running into issues with normalising the excited states. There doesn't seem to be a true convention for the ladder ...
0
votes
1answer
368 views

Expectation value in the ground state of simple harmonic oscillator

Motivated by a problem in chapter 2 of Sakurai's book Modern Quantum Mechanics, I'm interested in confirming something about the simple harmonic oscillator in quantum mechanics, I have found that the ...
0
votes
2answers
108 views

Expectation values in harmonic ocsillators

I am currently trying to understand the impact of lowering and raising operators on the expectation values of certain operators in a Harmonic Oscillator when the energy eigenstate has been stated as $|...
0
votes
2answers
803 views

2D harmonic oscillator: polar versus Cartesian eigenstates

Consider the usual two-dimensional harmonic oscillator (2D HO) with the Hamiltonian $$ H = -\frac{1}{2}\nabla_x^2 + \frac{1}{2}x^2 -\frac{1}{2}\nabla_y^2 + \frac{1}{2}y^2. $$ In Cartesian coordinates, ...
0
votes
2answers
486 views

Harmonic oscillator's lowering operator acting on state bra

For harmonic oscillator in quantum mechanics we have a lowering operator ($\hat a$) which it's action on state ket is: $$\hat a\;|n\rangle=\sqrt n \;|n-1\rangle$$ Is following relation true for it's ...
4
votes
0answers
1k views

Position and momentum eigenstates in terms of creation and annihilation operator? [closed]

Consider a simple harmonic oscillator; the position operator is $\hat{x}=(a^\dagger+a)/\sqrt{2}$ and the momentum operator is $\hat{p}=-i(a-a^\dagger)/\sqrt{2}$. One may verify that the eigenstates ...
1
vote
2answers
541 views

Meaning of Harmonic oscilator correlation function In Quantum Mechanics

What is the physical interpretation of the quantity $$\xi(t)=\langle 0|x(t)x(0)|0\rangle=\frac{\hbar}{2m\omega}e^{-i\omega t}$$ where $|0\rangle$ is the ground state of the harmonic oscillator? I know ...
4
votes
3answers
1k views

Coherent states of harmonic oscillator

Why are coherent states of the harmonic oscillator called coherent? Coherent in what sense? Why are these states so special/useful? From Wikipedia: In physics, two wave sources are perfectly ...
5
votes
1answer
899 views

Creating a QM state of definite position in Fock space

I'm wondering if somebody could help me to finish a simple calculation. Let me first provide motivation for the question below: I would like to write a QM amplitude in the 'QFT-style', as $$\langle \...
2
votes
1answer
96 views

Coherent state calculation of $\left<\lambda|x^2|\lambda\right>$, getting two different answers?

Getting a strange result here. I am studying the Quantum Harmonic oscillator and dealing with coherent states. I know that I can expand the position operator as $x = \sqrt{ \frac{\hbar}{2 m \omega} } ...
1
vote
0answers
41 views

Normalization in the one-loop vacuum amplitude (partiton function) of the bosonic string

This question stems from the string theory book by Ibanez and Uranga page 83, [link to excerpt], and Polchinksi [page 209 of volume 1]. The partition function has a piece $Z_{i,n}$ which Ibanez and ...
0
votes
1answer
115 views

Analysis of Harmonic Oscillator [duplicate]

In griffiths, Why was the lowest chosen like that while in the ladder analogy, the lower energies are like $a_-^n\psi$? That is, why didnt he chose the lowest rung to be $$a_-^n\psi$$
5
votes
2answers
400 views

Quantization of an Hamiltonian

My question is the following : I am studying a 1D harmonic oscillator chain. My classical hamiltonian contains terms such as $U_n$ where $U_n=x_n-x_n^0$, it represents the position away from ...
1
vote
3answers
266 views

Finding the Expectation value of harmonic oscillator [closed]

What is the expectation value for $e^{{\alpha}a^{\dagger}}e^{-\alpha^*{a}}$ of any two states of harmonic oscillator (let say $|n\rangle$ and $|m\rangle$) given below, $$\langle n|e^{{\alpha}a^{\...
1
vote
2answers
599 views

Completeness relation for coherent states of the quantum harmonic oscillator

For the Quantum harmonic oscillator with energy eigenstates $|n\rangle$ one defines a coherent state for every complex number $z$ by setting (note that the normalization varies across the literature) $...
3
votes
2answers
854 views

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 ...
1
vote
2answers
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 ...
0
votes
2answers
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.
3
votes
1answer
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 ...
0
votes
1answer
508 views

Eigenstates of sum of creation and annihilation operators

Does the operator $a+a^\dagger$ have eigenstates? If yes, what are they?
3
votes
0answers
178 views

Creating an arbitrary state of the quantum simple harmonic oscillator

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\...
-3
votes
1answer
645 views

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 $\vert 0\rangle$?...