All Questions
Tagged with harmonic-oscillator hilbert-space
167 questions
0
votes
0
answers
50
views
Can all solutions from any possible dynamical systems be formulated as combinations of harmonic oscillators?
In my understanding, all physical oscillators are unit vectors of a Hilbert space that is represented from the group $SU(3)\times SU(2)\times U(1)$. Every dynamical system operates under this group (...
- 929
1
vote
0
answers
36
views
Why Is There No Oscillator Representation for Operators in Planar ${\cal N}=4$ SYM Theory?
I'm studying the planar ${\cal N}=4$ Super Yang-Mills (SYM) theory and I'm curious about the representations of its operators, specifically the Hamiltonian and the dilatation operator. In many quantum ...
- 43
0
votes
1
answer
142
views
How is the quantum harmonic oscillator related to Fock states?
The question is basically in the title.
From what I understand, in the Fock state there is a certain number of particles in each energy level. The creation/annihilation operators create or destroy a ...
- 329
10
votes
3
answers
1k
views
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 $$\hat{H} = -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial ...
- 221
0
votes
0
answers
31
views
How can you have position basis and energy basis? [duplicate]
In Quantum Mechanics, my understanding is that we have a Hilbert space.
If we to model a particle in space we consider the space defined by the basis
$$|x\rangle$$
for each $x \in \mathbb{R}$
We then ...
7
votes
1
answer
211
views
How to get the factor of $n^{-27/4}$ in number of open string states from the calculation in GSW's book?
In section 2.3.5 of Green, Schwarz, Witten's book on string theory (volume-1) pp. 116-118, the objective is to calculate an Asymptotic Formula for Level Densities $d_n$ for open bosonic string theory. ...
- 822
1
vote
2
answers
69
views
Coherent creation operator: unitary or not?
In Quantum Mechanics, for coherent states $|z\rangle$ it can be prooved that if $|0\rangle$ is the vacuum state for an harmonic oscillator, therefore:
\begin{equation}
|z\rangle=e^{za^{\dagger}-z^*a}|...
1
vote
1
answer
110
views
When can we write $B=A^{\dagger}A?$
Consider ladder operators $a$ and $a^{\dagger}$ of a quantum harmonic oscillator. Now again say we define an operator $B=B(a,a^{\dagger})$ which is hermitian i.e., $B=B^{\dagger}$. The question is ...
- 67
0
votes
2
answers
99
views
Why does proportionality in eigenstates of the quantum harmonic oscillator not lead to degeneracy? [duplicate]
While I acknowledge that this topic has been discussed extensively, and I've read numerous similar questions along with their respective answers, I am still struggling to comprehend why all the ...
- 81
0
votes
2
answers
140
views
Finding the Expectation Value of Position for Generic $|n\rangle$ State
For a homework problem I had to write the position and impulse operators in terms of the creation and annihilation operators. The result is as follows:
$$\hat{x} = \sqrt{\frac{\hbar}{2 m \omega }} (a +...
- 1
3
votes
1
answer
95
views
Relation beteen harmonic polynomials and spherical harmonics
Considering a spinless particle moving in the 3D space and described by the wavefunction $$\psi(\vec{x})=(x^3+y^3)\frac{e^\frac{-r}{2\lambda}}{r^2}$$ where $\lambda$ is a lengthscale.
I want to know ...
- 336
0
votes
0
answers
82
views
2D Quantum Hamonic oscillator in magnetic field with a shiftted position
Background
Consider a hole in a 2D parabolic potential in a magnetic field which is generate by the following gauge:
$$
\vec{A} = \left( - \frac{B_z y}{2}, \frac{B_z x}{2},0\right)
$$
Our quantum ...
1
vote
1
answer
142
views
Quantum state of a harmonic oscillator given energy and probability [closed]
Hi I've been going through exercises without solution, so I would like to ask for some feedback if I am solving this right:
A one-dimensional quantum harmonic oscillator with mass $m$ and frequency $\...
- 39
3
votes
1
answer
474
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
1
answer
147
views
Applying the position operator several times to a harmonic oscillator state $\hat x^m |n\rangle =$ ______? [duplicate]
The position operator can be expressed in terms of the harmonic oscillator ladder operators
$$\hat x = \hat a + \hat a^\dagger,$$
in natural units. Therefore we have
$$\hat x |n\rangle = \frac{\hat a +...
- 2,785
1
vote
2
answers
644
views
Quantum mechanical harmonic oscillator - where does the number operator come from?
I have recently been learning about the quantum harmonic oscillator and how it is described in the language of ladder operators. At the moment the logic behind the number operator seems incomplete to ...
- 13
4
votes
2
answers
325
views
Meaning of "the symmetry group of an $N$-dimensional quantum isotropic oscillator is $U(N)$"
Symmetry of this system has been discussed here but I'm still confused.
Consider a $N$-dimensional isotropic harmonic oscillator, with hamiltonian
$$H = \hbar \omega \left(a^\dagger_i a_i + \frac{N}{2}...
2
votes
2
answers
392
views
Is linear momentum quantized in quantum harmonic oscillator
I'm self-studying QM and have a basic question on quantum harmonic oscillator. The Hamilton is certainly quantized under this model, that is $E_n=(n+1/2)\hbar \omega$, for $n=0,1,2,...$. But is linear ...
- 107
2
votes
0
answers
99
views
Overlap of Vacuum States (Problem 2.4 of Modern Quantum Field Theory by Thomas Banks) [closed]
Problem
Compute the overlap of the ground-state wave functions of a harmonic oscillator with two different frequencies. A free-bosonic field theory is just a collection of oscillators. Use your ...
-1
votes
3
answers
175
views
Harmonic Oscillator Eigenket Notation
I'm reading the $3^{\mathrm{rd}}$ edition of Sakurai and Napolitano's Modern Quantum Mechanics, and I have a brief question about the notation used to describe the eigenstates of the harmonic ...
- 385
2
votes
2
answers
319
views
Understanding how to terminate recurrence relations in quantum SHO
In the coordinate representation solution to the quantum SHO (the solution via differential equations rather than Dirac's "trick") we ultimately work out that our eigenfunction solutions are ...
- 1,261
0
votes
1
answer
88
views
Why is the commutator of ladder operators non-zero?
Griffiths states that the "ladder" of stationary states for a harmonic oscillator should be unique. That should mean that for one particular energy level, there exists only one energy state. ...
- 67
3
votes
1
answer
94
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
$$
H a |{n}\rangle = (E_n - \hbar \omega_0)a|{n}\rangle,
$$
$$
H a^\...
- 121
0
votes
1
answer
421
views
Calculating $\hat{x}^2$ and $\hat{p}^2$ - harmonic oscillator matrix form [closed]
In harmonic oscillator, we can write $\hat{x}$ and $\hat{p}$ as (I obtained the $\hat{x}$ and $\hat{p}$ by using matrix form of the ladder operators) ;
$$\hat{x} = \sqrt{\frac{\hbar}{2m\omega}}\begin{...
- 3,160
2
votes
1
answer
613
views
Is the average position for the ground state of a 1D simple harmonic oscillator zero? [closed]
My textbook claims the average position of the $n$-th state of 1D simple harmonic oscillator (SHO) is zero, which means
$$
\def\bra #1{\langle #1 |}
\def\ket #1{| #1 \rangle}
\def\braket #1{\langle #1 ...
- 277
1
vote
1
answer
174
views
Vanishing zero point energy in harmonic oscillator
In classical mechanics, adding a constant to the potential changes nothing. In quantum mechanics, this just shifts the energy and multiples the wavefunction with a phase term.
But now suppose I use ...
- 2,020
0
votes
1
answer
90
views
How does an operator act when taking an expectation value? [closed]
So I am reading this book where they say this:
My understanding of how operators work is that :
$$\langle n|\hat{a}|\alpha \rangle = \langle n|\hat{a}\alpha \rangle = \langle n\hat{a}^\dagger |\alpha ...
- 534
0
votes
1
answer
86
views
A doubt regarding Quantum Harmonic Oscillator
Classically when we solve Newton's equation for $V=\frac{1}{2}m\omega^2x^2$ we get two linearly independent solutions (for $\omega\not=0$): $Ae^{\omega t}$ & $Be^{-\omega t}$, their linear ...
user292464
2
votes
0
answers
62
views
Could one call eigenstates of $ \hat{a} = \hat{x} + i\hat{p}$ coherent states for other potentials than the harmonic oscillator?
Let's say I look at the quantum system of a particle in one dimension, subject to any other potential than the one of the harmonic oscillator, and I define $\hat{a}$ as stated above. I would find the ...
- 6,970
2
votes
1
answer
117
views
Zero frequency quantum oscillator rigorously
Problem description
Consider quantum oscillator with $ \omega = 0 $. In other words, we have
$$
\hat{H} = \hat{p}^2,
$$
where $\hat{x}, \, \hat{p}$ are the usual coordinate and momentum operators with ...
2
votes
2
answers
715
views
Square root of number operator for quantum harmonic oscillator
Let $a$, $a^{\dagger}$ denote the standard annihilation and creation operators for the quantum harmonic oscillator, with $[a, a^{\dagger}] = \mathbb{I}$. The number operator is then defined as $a^{\...
- 397
1
vote
1
answer
79
views
Why is there only one eigenket per eigenvalue of the number operator? [duplicate]
Let's "define" (I put quotes since it's not a definition, but just requiring a property) the operator $a$ such that:
$$[a,a^\dagger]=1$$
then
$$n=a^\dagger a$$
No other assumptions are made ...
- 43
0
votes
1
answer
192
views
What is the justification of imposing the commutation relation between $a$ and $a^\dagger$ in the quantized electromagnetic field? [duplicate]
singles out one mode, one quantum object, from the rest of the world. This object turns out to be a harmonic oscillator described by the annihilation operator $\hat{a}$. The operator $\hat{a}$ stands ...
3
votes
1
answer
976
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 '...
- 523
2
votes
0
answers
504
views
Proof of Wick's theorem for general Gaussian states
For boson modes $a^\dagger_i, a_i$, consider the density matrix which is an exponential of quadratic operators in $a^\dagger_i$ and $a_i$:
$$\rho = e^{-H_{ij} a^\dagger_i a_j + (K_{ij} a^\dagger_i a^\...
- 1,101
2
votes
0
answers
115
views
Hilbert space of a diatomic molecule
In molecular quantum mechanics, it is very common to model a diatomic molecule as a two-level harmonic oscillator with vibrational levels lying within the electronic states:
In most of the textbooks ...
- 61
0
votes
1
answer
61
views
$n$-number of creation operators on the ground state [closed]
I simply want to prove the following:
for the given state, $|n\rangle = \frac{1}{\sqrt{n!}}(a^\dagger)^n|0\rangle$, show that this satisfies $\hat{N}|n\rangle = n|n\rangle$ given $\hat{N} = \hat{a}^\...
- 923
1
vote
0
answers
61
views
Can $A_{nm} = |n\rangle \langle m|$ be written in terms of boson operators $b, b^\dagger$? [duplicate]
I am curious whether all operators can be written as a linear combination of product of boson operators $b, b^\dagger$.
More precisely, consider the single harmonic oscillator Hilbert space $H$, whose ...
- 1,101
5
votes
3
answers
250
views
How to express $|m\rangle\langle n|$ in terms of ladder operators?
Let us consider the Hamiltonian of a single harmonic oscillator, which is expressed in terms of creation/annihilation operators as
$H=\hbar \omega (a^{\dagger}a+1/2)$. The eigenstates of this ...
- 462
1
vote
0
answers
90
views
Mathematical issue regarding a variant of Quantum Harmonic Oscillator
Suppose I have a potential of the form $$V(x) = \begin{cases} \frac{1}{2}kx^2 &\text{ if }x>0 \\ {\infty} &\text{ if }x<0 \end{cases} $$
We know for the usual HO one introduces $a$ and $...
- 395
0
votes
2
answers
461
views
Eigenvalue Equation : $ A |\psi\rangle = 0 |\phi\rangle$?
A typical eigenvalue equation goes like: $ A |\psi\rangle = e\: |\psi\rangle$, where $|\psi\rangle$ is an eigenstate for operator $A$ with eigenvalue $e$.
Suppose that $e=0$ in the above equation, ...
- 698
0
votes
1
answer
474
views
Quantum Harmonic Oscillator eigenfunction
I'm trying to understand why in quantum harmonic oscillator when finding ground state eigenfunction we don't use $a^\dagger$.
For a simple harmonic oscillator the Hamiltonian is given by $$H=\hbar\...
- 340
2
votes
0
answers
220
views
Lindblad form for a Damped harmonic Oscillator
I'm considering a Lindblad-like master equation for a damped harmonic oscillator
$$
\frac{d\rho(t)}{dt} = -i[H, \rho(t)]+\sum_{n,m=1}^2 h_{nm}[A_n \rho(t) A_m^\dagger
- \frac12 \lbrace A_m^\...
- 475
29
votes
3
answers
3k
views
Do states with infinite average energy make sense?
Do states with infinite average energy make sense?
For the sake of concreteness consider a harmonic oscillator with the Hamiltonian $H=a^\dagger a$ and eigenstates $H|n\rangle=n|n\rangle$, $\langle n|...
- 1,594
1
vote
1
answer
144
views
Uncertainty in eigenstate of $Q=\mu a+\nu a^\dagger$
Consider a particle in simple harmonic oscillator potential. Suppose the eigenstate of the operator
$$Q=\mu a+\nu a^\dagger$$
with eigenvalue $\alpha$ where $\mu,\nu $ and $\alpha$ are three complex ...
- 12.1k
0
votes
2
answers
725
views
Quantum Harmonic Oscillator With a Sudden Change in Electric Field
Consider the following hamiltonian:
$$H=\begin{cases}
\frac{p^2}{2m}+\frac{1}{2}m\omega_0^2x^2,&\text{ for }t<0\\
\frac{p^2}{2m}+\frac{1}{2}m\omega_0^2x^2-q\epsilon_0x,&\text{ for }...
- 269
0
votes
1
answer
192
views
How does the wavefunction look like for inverted oscillator potential? [duplicate]
Suppose the inverted harmonic oscillator potential
$$H=\frac{p^2}{2m}-\frac{1}{2}m\omega^2x^2$$
I'm looking for a form of solution for the case when $E<0$. It's clear that a scattering solution ...
- 12.1k
1
vote
4
answers
771
views
Hamiltonian of quantum harmonic oscillator - how it affects the dynamics of a system?
The quantum harmonic oscillator can be described with the creation and annihilation operators of its eigen states:
$$H=\hbar \omega\left(a^+a+\frac{1}{2}\right) \, .$$
Which possesses the following ...
- 239
4
votes
3
answers
925
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 ...
- 1,461
3
votes
2
answers
277
views
Why is $\hat{a}^{\dagger 2}\hat{a}^2|\alpha\rangle=|\alpha|^4|\alpha\rangle$ for $|\alpha\rangle$ a coherent state of a QM harmonic oscillator?
I'm currently trying to understand the following:
Consider a quantum harmonic oscillator with a coherent state $|\alpha\rangle$. Show that $\langle E^2\rangle=\hbar\omega (|\alpha|^4+2|\alpha|^2+\frac{...
- 239