All Questions
Tagged with harmonic-oscillator hilbert-space
162
questions
57
votes
4
answers
6k
views
Hilbert space of harmonic oscillator: Countable vs uncountable?
Hm, this just occurred to me while answering another question:
If I write the Hamiltonian for a harmonic oscillator as
$$H = \frac{p^2}{2m} + \frac{1}{2} m \omega^2 x^2$$
then wouldn't one set of ...
- 14.8k
29
votes
3
answers
3k
views
Do states with infinite average energy make sense?
Do states with infinite average energy make sense?
For the sake of concreteness consider a harmonic oscillator with the Hamiltonian $H=a^\dagger a$ and eigenstates $H|n\rangle=n|n\rangle$, $\langle n|...
- 1,556
26
votes
2
answers
9k
views
Proof that the one-dimensional simple harmonic oscillator is non-degenerate?
The standard treatment of the one-dimensional quantum simple harmonic oscillator (SHO) using the raising and lowering operators arrives at the countable basis of eigenstates $\{\vert n \rangle\}_{n = ...
- 397
12
votes
3
answers
10k
views
Why Don't the Ladder Operators Commute?
I have two problems with ladder operators.
The first is that I feel they should somehow result in measurable things. The asymmetry of applying the plus operator versus the minus operator is very ...
user24082
10
votes
2
answers
12k
views
Eigenstates of a shifted harmonic oscillator
Let's say I have a quantum harmonic oscillator $H = \omega a^\dagger a$, where $a^\dagger$ is the raising operator and $a$ is the lowering operator and $H |n\rangle = \omega n |n\rangle$.
Now assume ...
- 1,022
10
votes
1
answer
3k
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
$$\langle \...
- 2,852
9
votes
4
answers
871
views
Ladder operator identity for $\langle n | (a+a^\dagger)^k | m \rangle$
I would like to know if there is a convenient identity (and what it is) for
$$\langle n | (a+a^\dagger)^k | m \rangle$$
where $| n \rangle, \, | m \rangle$ are energy eigenstates of a simple ...
- 1,020
9
votes
2
answers
944
views
Do the ladder operators $a$ and $a^\dagger$ form a complete algebra basis?
It is easy to construct any operator (in continuous variables) using the set of operators $$\{|\ell\rangle\langle m |\},$$ where $l$ and $m$ are integers and the operators are represented in the Fock ...
9
votes
1
answer
805
views
Proof that $\exp(-\beta H)$ is a trace-class operator for the harmonic oscillator
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&...
- 507
7
votes
3
answers
4k
views
Why is the action of lowering operator on the ground state of a harmonic oscillator to give a 0 wave function?
In quantum mechanics of the harmonic oscillator, when we use the operator method to find out the solutions, we find that the action of $\hat{a}$ is to lower the energy of a state by $\hbar\omega$ and ...
- 773
7
votes
1
answer
164
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How to get the factor of $n^{-27/4}$ in number of open string states from the calculation in GSW's book?
In section 2.3.5 of Green, Schwarz, Witten's book on string theory (volume-1) pp. 116-118, the objective is to calculate an Asymptotic Formula for Level Densities $d_n$ for open bosonic string theory. ...
- 414
6
votes
2
answers
14k
views
How To Use Ladder Operators?
I'm studying for a test in quantum mechanics and I'm having a hard time understanding how to use ladder operators. There are no examples in my text book, only definitions that I can't understand how ...
- 463
6
votes
1
answer
609
views
Orbital angular momentum as sum of harmonic oscillators
On section 7.3 of Ballentine's "Quantum Mechanics: A Modern Development" there is a really nice argument on why the eigenvalues of the total angular momentum operator must be integer, cf. e....
- 93
6
votes
2
answers
612
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 ...
- 323
5
votes
1
answer
2k
views
The issue on existence of inverse operations of $a$ and $a^{\dagger}$
I have asked a question at math.stackexchange that have a physical meaning.
My assumption: Suppose $a$ and $a^\dagger$ is Hermitian adjoint operators and $[a,a^\dagger]=1$. I want to prove that ...
- 1,014
5
votes
2
answers
2k
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 ...
- 8,409
5
votes
2
answers
970
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 \...
- 53
5
votes
2
answers
1k
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 ...
- 1,350
5
votes
0
answers
3k
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 ...
- 2,071
4
votes
1
answer
771
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),
$$
with eigenstates $|1\rangle,|2\rangle,\ldots$
Many textbooks define the annihilation operator to ...
- 1,315
4
votes
3
answers
222
views
How to express $|m\rangle\langle n|$ in terms of ladder operators?
Let us consider the Hamiltonian of a single harmonic oscillator, which is expressed in terms of creation/annihilation operators as
$H=\hbar \omega (a^{\dagger}a+1/2)$. The eigenstates of this ...
- 412
4
votes
3
answers
832
views
Is the zero point energy of this system zero?
Consider the following Hamiltonian: $$\hat H=\frac{\hbar\omega}{2}(\hat x^2+\hat p^2)-\frac{\hbar\omega}{2}\hat 1 =\frac{\hbar\omega}{2}(\hat x^2+\hat p^2-\hat 1 )$$
After defining annihilation and ...
- 1,431
4
votes
1
answer
268
views
On the two different solution approaches of the quantum harmonic oscillator
The Hamiltonian for the harmonic oscillator (with $\hbar = m = 1$) is given by:
$$\hat{H} = -\frac{1}{2}\frac{d^{2}}{dx^{2}} + \frac{1}{2}\omega^{2}x^{2}$$
This is assumed to be an operator on $\...
- 1,123
4
votes
3
answers
3k
views
What is the Hamiltonian in the "energy basis" for a simple harmonic oscillator?
My textbook says that for a simple harmonic oscillator the Hamiltonian can be expressed in the "energy basis" in this way:
$$\hat H=\hbar\omega\bigg(\hat a^{\dagger}\hat a + {1\over 2}\bigg).$$
I ...
- 805
4
votes
1
answer
343
views
Individual particle states in Fock space
I am currently learning QFT, and after watching the wonderful lectures by Leonard Susskind (https://theoreticalminimum.com/courses/advanced-quantum-mechanics/2013/fall), I am still struggling to see ...
- 343
4
votes
2
answers
2k
views
Meaning of Harmonic oscilator correlation function In Quantum Mechanics
What is the physical interpretation of the quantity $$\xi(t)=\langle 0|x(t)x(0)|0\rangle=\frac{\hbar}{2m\omega}e^{-i\omega t}$$ where $|0\rangle$ is the ground state of the harmonic oscillator? I know ...
- 26.4k
3
votes
4
answers
810
views
Hamiltonian of quantum harmonic oscillator with $\psi(x)=\delta(x)$: comparison to classical mechanics
I was just reading the question Why can't $\psi(x)=\delta(x)$ in the case of a harmonic oscillator? The accepted answer says that $\psi(x)=\delta(x)$ is a mathematically valid state, though it's not ...
user191954
3
votes
2
answers
274
views
Meaning of "the symmetry group of an $N$-dimensional quantum isotropic oscillator is $U(N)$"
Symmetry of this system has been discussed here but I'm still confused.
Consider a $N$-dimensional isotropic harmonic oscillator, with hamiltonian
$$H = \hbar \omega \left(a^\dagger_i a_i + \frac{N}{2}...
3
votes
1
answer
420
views
Do Hermite polynomials imply a weight for quantum harmonic oscillator wavefunctions?
I know that solutions of quantum harmonic oscillator can be expressed in the form of Hermite polynomials.
But I recently came to know that Hermite polynomials are actually orthogonal polynomials ...
- 431
3
votes
1
answer
266
views
Why is the degeneracy of the 3D isotropic quantum harmonic oscillator finite?
This is a more conceptual question.
When we take the isotropic harmonic oscillator:
$$V = \frac{1}{2}m\omega^2(x^2+y^2+z^2)$$
the eigenvalue equation solves to:
$$\phi_E=\phi_l(x)\phi_m(y)\phi_n(z)$$
...
- 105
3
votes
3
answers
233
views
Procedure to cut an Harmonic oscillator to two first level to obtain a qubit
Let us consider a (quantum) Harmonic oscillator:
$$H=\frac{p^2}{2m}+\frac{1}{2} m \omega^2 x^2$$
Using the annihilation/creation operators defined as:
$$a=\sqrt{\frac{\hbar}{2 m \omega}}(x+\frac{i}{m \...
- 1,350
3
votes
3
answers
551
views
When Is It Appropriate To Use The Ladder Operator Method in Quantum Mechanics?
I'm trying to understand when it is intuitively obvious that the ladder method would be best used to tackle a problem in quantum mechanics.
- 401
3
votes
1
answer
559
views
Confusion about modes and quantum field theory
I'm learning quantum field theory from P&S and Srednicki. I'm having a lot of difficulties understanding the concept of a momentum state. In particular, I'm confused about how to interpret the ...
user299467
3
votes
1
answer
331
views
Why can't $\psi(x) = \delta(x)$ in the case of Harmonic oscillator?
In the analysis of Harmonic Oscillator, it is claimed that $\langle\hat H\rangle$ cannot be zero, why is it so?
I mean $\hat H = \frac{ \hat p^2 }{2m } + \frac12 k \hat x^2$, and
$$\left<x^2\...
- 2,263
3
votes
1
answer
699
views
Relation between the bosonic harmonic oscillator and the 'ordinary' harmonic oscillator
What exactly is the connection between the Harmonic oscillator that we study in Quantum Mechanics I, and the bosonic (and fermionic) harmonic oscillator ?
In some sources, it is claimed that my '...
- 483
3
votes
1
answer
769
views
Why does the virial theorem of quantum mechanics hold for the quantum oscillator but not the infinite square well?
Why does the virial theorem of quantum mechanics hold for the quantum oscillator but not the infinite square well? The proof uses Ehrenfest's theorem, so I was wondering if it had something to do with ...
- 43
3
votes
2
answers
3k
views
The harmonic oscillator - ladder operators
Reading from Griffiths. I have got two questions.
First, the halmiltonian operator that used to find the energy eigenvalue in only harmonic oscillator is:
$$H={\hbar}w(a_-a_+-\frac{1}{2})$$ and $$H={\...
- 495
3
votes
1
answer
89
views
Proof that the QHO annihilation operator has nullity of 1
In a traditional analysis of a quantum harmonic oscillator (QHO), operators $a$ and $a^\dagger$ are introduced and it is shown that
$$
H a |{n}\rangle = (E_n - \hbar \omega_0)a|{n}\rangle,
$$
$$
H a^\...
- 121
3
votes
2
answers
774
views
Quantum Mechanics Basics: product space
Consider a coupled harmonic oscillator with their position given by $x_1$ and $x_2$. Say the normal coordinates $x_{\pm}={1\over\sqrt{2}} (x_1\pm x_2)$, in which the harmonic oscillators decouple, ...
- 163
3
votes
2
answers
397
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 ...
- 33
3
votes
2
answers
681
views
Harmonic Oscillator - Energy quantisation
The one-dimensional quantum HO can be solved in Schrodinger representation by getting Hermite Differential Equation
$$ \frac{d^2y}{dx^2} - 2x \frac{dy}{dx} + \lambda y = 0 $$
with solutions
$$ y(x) = ...
- 2,975
3
votes
2
answers
207
views
Question about lowest rung on Harmonic Oscillator
I'm reading Griffiths's Introduction to Quantum Mechanics 3rd ed textbook [1]. On p.43, the author explains:
What if I apply the lowering operator repeatedly? Eventually I’m going to reach
a state ...
- 441
3
votes
1
answer
557
views
State time evolution of a quantum harmonic oscillator with a Dirac-Delta function as an initial state [closed]
I have a question just like this Phys.SE problem here with a difference that our system is a harmonic oscillator (rather than a free particle). A particle with mass $m$ is connected to a string with ...
- 431
3
votes
0
answers
197
views
Creating an arbitrary state of the quantum simple harmonic oscillator
Suppose $\mathcal{B}=\{|0\rangle, |1\rangle, |2\rangle, ... \}$ is the energy eigen-basis of a quantum simple harmonic oscillator. I want to create the state
\begin{equation}
|\Psi\rangle = c_{0}|0\...
- 316
2
votes
3
answers
3k
views
How does one calculate the position eigenvalues of the matrix corresponding to the position operator?
The matrix representation corresponding to the position operator is:
$$x = \sqrt{\frac{\hbar}{2 m \omega}} \left[
\begin{array}{ccccc}
0 & \sqrt{1} & 0 & 0 & \cdots \\
\sqrt{1} & ...
2
votes
3
answers
557
views
What is the physical meaning of $a_{\vec{p}} \! \mid \! 0 \rangle$
$a^\dagger_{\vec{p}} \! \mid \! 0 \rangle = \mid \! p \rangle$ is interpreted as a creation of a particle with momentum $p$ from the vacuum. $a_{\vec{p}} \! \mid \! p \rangle = \mid \! 0 \rangle$ is ...
- 5,319
2
votes
3
answers
583
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 ...
- 110
2
votes
1
answer
228
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} } ...
- 3,659
2
votes
3
answers
18k
views
Degeneracy of 2 Dimensional Harmonic Oscillator
If we consider a particle in a 2 dimensional harmonic oscillator potential with Hamiltonian $$H = \frac{\mathbf{p}^2}{2m} + \frac{m w^2 \textbf{r}^2}{2}$$ it can be shown that the energy levels are ...
user154080
2
votes
1
answer
929
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 ...
user100411