What is Ladder operators: Definition and 72 Discussions
In linear algebra (and its application to quantum mechanics), a raising or lowering operator (collectively known as ladder operators) is an operator that increases or decreases the eigenvalue of another operator. In quantum mechanics, the raising operator is sometimes called the creation operator, and the lowering operator the annihilation operator. Well-known applications of ladder operators in quantum mechanics are in the formalisms of the quantum harmonic oscillator and angular momentum.
Hi Pfs
i have 2 matrix representations of SU(2) . each of them uses a up> and down basis (d> and u>
If i take their tensor product i will get 4*4 matrices with this basis:
d>d>,d>u>,u>d>,u>u>
these representation is the sum equal to the sum of the 0-representation , a singlet represertation with...
I've tried figuring out commutation relations between ##L_+## and various other operators and ##L^2## could've been A, but ##L_z, L^2## commute. Can someone help me out in figuring how to actually proceed from here?
First time posting in this part of the website, I apologize in advance if my formatting is off.
This isn't quite a homework question so much as me trying to reason through the work in a way that quickly makes sense in my head. I am posting in hopes that someone can tell me if my reasoning is...
I'm trying verify the proof of the sum rule for the one-dimensional harmonic oscillator:
$$\sum_l^\infty (E_l-E_n)\ | \langle l \ |p| \ n \rangle |^2 = \frac {mh^2w^2}{2} $$
The exercise explicitly says to use laddle operators and to express $p$ with
$$b=\sqrt{\frac {mw}{2 \hbar}}-\frac...
The previous part was to show that ##a_+ \psi_n = i\sqrt{(n+1)\hbar \omega} \psi_{n+1}##, which I just did by looking at$$\int |a_+ \psi_n|^2 dx = \int \psi_n^* (a_{-} a_+ \psi_n) dx = E+\frac{1}{2}\hbar \omega = \hbar \omega(n+1)$$so the constant of proportionality between ##a_+ \psi_n## and...
In theory, does an algebraic expression exist for the ground state of the Klein Gordon equation with \phi^4 interactions in the same way an algebraic expression exists for the simple harmonic oscillator ground state wavefunction in Q.M.? Is it just that it hasn't been found yet or is it...
I know that ahat_+ = 1/sqrt((2*m*h_bar*w)) * (mw(xhat)+i(phat)) and ahat_- = 1/sqrt((2*m*h_bar*w)) * (mw(xhat)-i(phat)). But I'm not sure what (ahat_+)^+ could be.
a. ##L_{+}^{\dagger}=(L_x+iL_y)^{\dagger}=L_x-iL_y=L_{-}##
b.##[L_{+},L_{-}]=[L_x+iL_y,L_x-iL_y]=(L_x+iL_y)(L_x-iL_y)-(L_x-iL_y)(L_x+iL_y)=##
##=L_x^2-iL_xL_y+iL_yL_x+L_y^2-(L_x^2+iL_xL_y-iL_yL_x-L_y^2)##
##=L_x^2-iL_xL_y+iL_yL_x+L_y^2-L_x^2-iL_xL_y+iL_yL_x+L_y^2##...
I've been reading this book, in which the author expresses the vacuum projection operator ##\vert 0\rangle\langle 0\vert## in terms of the number operator ##\hat{N}=\hat{a}^{\dagger}\hat{a}##, where ##\hat{a}^{\dagger}## and ##\hat{a}## are the usual creation and annihilation operators...
Hi! I am working on homework and came across this problem:
<n|X5|n>
I know X = ((ħ/(2mω))1/2 (a + a+))
And if I raise X to the 5th, its becomes X5 = ((ħ/(2mω))5/2 (a + a+)5)
What I'm wondering is, is there anyway to be able to solve this without going through all of the iterations the...
Homework Statement
I am trying to improve my understanding of the Clebsch-Gordan coefficients. I am looking at page 5 of the following document https://courses.physics.illinois.edu/phys570/fa2013/chapter3.pdf
Homework Equations
I have derived the result for the I = 3/2 quadruplet but am...
Hello! I read in many places the derivation of the representation for SU(2) using ladder operators and in all of the places they say that, due to the fact that we are looking for a finite dimensional representation, the ladder must end at a point, hence why we have an eigenvector of ##L_3##...
1. The problem statement
I want to write the angular momentum operator ##L## for a 2-dimensional harmonic oscillator, in terms of its ladder operators, ##a_x##, ##a_y##, ##a_x^\dagger## & ##a_y^\dagger##, and then prove that this commutes with its Hamiltonian.
The Attempt at a Solution
I get...
Homework Statement
The spherical harmonic, Ym,l(θ,φ) is given by:
Y2,3(θ,φ) = √((105/32π))*sin2θcosθe2iφ
1) Use the ladder operator, L+ = +ħeiφ(∂/∂θ+icotθ∂/∂φ) to evaluate L+Y2,3(θ,φ)
2) Use the result in 1) to calculate Y3,3(θ,φ)
Homework Equations
L+Ym,l(θ,φ)=Am,lYm+1,l(θ,φ)...
A fictitious system having three degenerate angular momentum states with ##\ell=1## is described by the Hamiltonian \hat H=\alpha (\hat L^2_++\hat L^2_-) where ##\alpha## is some positive constant. How to find the energy eigenvalues of ##\hat H##?
Homework Statement
STATEMENT
##\hat{H}=\int \frac{d^3k}{(2\pi)^2}w_k(\hat{a^+(k)}\hat{a(k)} + \hat{b^{+}(k)}\hat{b(k)})##
where ##w_k=\sqrt{{k}.{k}+m^2}##
The only non vanishing commutation relations of the creation and annihilation operators are:
## [\alpha(k),\alpha^{+}(p)] =(2\pi)^3...
Homework Statement
Consider the free real scalar field \phi(x) satisfying the Klein-Gordon equation, write the Hamiltonian in terms of the creation/annihilation operators.
Homework Equations
Possibly the definition of the free real scalar field in terms of creation/annihilation operators...
I am getting started with QFT and I'm having a hard time to understand the quantization procedure for the simples field: the scalar, massless and real Klein-Gordon field.
The approach I'm currently studying is that by Matthew Schwartz. In his QFT book he first solves the classical KG equation...
S. Weinberg says in his book, "The Quantum Theory of Fields Volume I", that
Since electrons carry a charge, we would not like to mix annihilation and creation operators, so we might try to write the field as $$\psi(x)=\sum_{k}u_k (x)e^{-i\omega_k t}a_k$$
where ##u_k (x)e^{-i\omega_k t}## are a...
The ladder operators of a simple harmonic oscillator which obey
$$[H,a^{\dagger}]=\hbar\omega\ a^{\dagger}$$.
---
I would like to see a proof of the relation
$$\exp(-iHt)\exp(a^{\dagger})\exp(iHt)|0\rangle=\exp(a^{\dagger}e^{-i\omega t})|0\rangle\exp(i\omega t/2).$$
Thoughts?
Homework Statement
Obtain the matrix representation of the ladder operators ##J_{\pm}##.
Homework Equations
Remark that ##J_{\pm} | jm \rangle = N_{\pm}| jm \pm 1 \rangle##
The Attempt at a Solution
[/B]
The textbook states ##|N_{\pm}|^2=\langle jm | J_{\pm}^\dagger J_{\pm} | jm \rangle##...
Hi, guys.
This is actually a question about quantum mechanics, but since the context in which it appeared is particle physics, I'll post it here.
On Thompson's book (page 227, equation (9.32)), we have
$$T_+ |d\bar{u}\rangle = |u\bar{u}\rangle - |d\bar{d}\rangle$$
But I thought...
I would like to show that the commutation relations ##[a_{\vec{p}},a_{\vec{q}}]=[a_{\vec{p}}^{\dagger},a_{\vec{q}}^{\dagger}]=0## and ##[a_{\vec{p}},a_{\vec{q}}^{\dagger}]=(2\pi)^{3}\delta^{(3)}(\vec{p}-\vec{q})## imply the commutation relations...
Say I apply a raising operator to the spin state |2,-1>, then by using the the equation
S+|s,ms> = ћ*sqrt(s(s+1) - ms(ms+1))|s,ms+1>
I get,
S+|2,-1> = sqrt(6)ћ|2,0>
Does this correspond to a physical eigenvalue or should I disregard it and only take states with integer multiples of ћ as...
Due to the definition of spin-up (in my project ),
\begin{eqnarray}
\sigma_+ =
\begin{bmatrix}
0 & 2 \\
0 & 0 \\
\end{bmatrix}
\end{eqnarray}
as opposed to
\begin{eqnarray}
\sigma_+ =
\begin{bmatrix}
0 & 1 \\
0 & 0 \\
\end{bmatrix}
\end{eqnarray}
and the annihilation operator is...
Homework Statement
Consider a linear harmonic oscillator with the solution defined by the ladder operators a and a†. Use the number basis |n⟩ to do the following.
a) Construct a linear combination of |0⟩ and |1⟩ to form a state |ψ⟩ such that ⟨ψ|X|ψ⟩ is as large as
possible.
b) Suppose that...
The quantum Klein-Gordon field ##\phi({\bf{x}})## and its momentum density ##\pi({\bf{x}})## are given in Fourier space by
##\phi({\bf{x}}) = \int \frac{d^{3}p}{(2\pi)^{3}} \frac{1}{\sqrt{2 \omega_{{\bf{p}}}}} \big( a_{{\bf{p}}} e^{i{\bf{p}} \cdot {\bf{x}}} + a^{\dagger}_{{\bf{p}}}...
Please I need your help in such problems..
in terms of ladder operators to simplify the calculation of matrix elements... calculate those
i) <u+2|P2|u>
ii) <u+1| X3|u>
If u is different in both sides, then the value is 0? is it right it is 0 fir both i and ii?
when exactly equals 0, please...
Homework Statement
Consider the following state constructed out of products of eigenstates of two individual angular momenta with ##j_1 = \frac{3}{2}## and ##j_2 = 1##:
$$
\begin{equation*}
\sqrt{\frac{3}{5}}|{\tiny\frac{3}{2}, -\frac{1}{2}}\rangle |{\tiny 1,-1}\rangle +...
Puting a minus in front of the momentum in the field expansion gives
##\phi \left( {\bf{x}} \right) = \int {{d^3}\tilde p} \left( {{a_{\bf{p}}}{e^{i{\bf{p}} \cdot {\bf{x}}}} + a_{\bf{p}}^ + {e^{ - i{\bf{p}} \cdot {\bf{x}}}}} \right){\rm{ }}\phi \left( {\bf{x}} \right) = \int {{d^3}\tilde...
Let a be a lowering operator and a† be a raising operator.
Prove that a((a†)^n) = n (a†)^(n-1)
Professor suggested to use induction method with formula:
((a†)(a) + [a,a†]) (a†)^(n-1)
But before start applying induction method, I would like to know where the given formula comes from. Someone...
Hello, I was just watching a youtube video deriving the equation for the Hamiltonian for the harmonic oscillator, and I am also following Griffiths explanation. I just got stuck at a part here, and was wondering if I could get some help understanding the next step (both the video and book...
I am referring to the section The Harmonic Oscillator in Griffiths's introductino to quantum mechanics (the older edition with the black cover). I understand how it all works, however there is a part that I am not sure about. How do we know when we apply a- or a+ (the ladder operators) to a...
When using Fourier's trick for determining the allowable energies for stationary states, Griffiths introduces the a+- operators. When factoring the Hamiltonian, the imaginary part is assigned to the momentum operator versus the position operator. Is there a reason for this? If :
a-+ = k(ip +...
Hello,
I am reading Griffiths Quantum Mechanics textbook, and am having some difficulty with a derivation on page 56. To me, there seems to be something logically wrong with his arguments, but I can not pin-point precisely what it is.
To provide you with a little background, Griffiths is...
Hello,
I am currently studying ladder operator for a simple harmonic operator as a method for generating the energy values. This seem like a simple algebra question I am asking so I do apologize but I just can't figure it out. Here are my operator definitions...
Homework Statement
For the SHO, find these commutators to their simplest form:
[a_{-}, a_{-}a_{+}]
[a_{+},a_{-}a_{+}]
[x,H]
[p,H]
Homework Equations
The Attempt at a Solution
I though this would be an easy problem but I am stuck on the first two parts. Here's what I did at first...
The total energy of a particle in a harmonic oscillator is found to be 5/2
~!. To change the energy,
if i applied the lowering operator 4 times and then the raising operator 1 times successively. What
will be the new total energy?
i want the calculation please
Reading through my QM text, I came across this short piece on ladder operators that is giving me trouble (see picture). What I am struggling with is how to get to equations 2 and 3 from equation 1.
Can someone point me in the right direction? Where does the i infront of the x go?
Hello,
I've read that Dirac introduced the idea of the creation and annihilation operators in the solution to the quantum harmonic oscillator problem, but can anyone tell me where he did this? In a paper, or maybe in a book?
I've had a little search online, but I've yet to discover...
What is the significance of the ladder operators eigenvalues as they act on the different magnetic quantum numbers, ml and ms to raise or lower their values?
How do their eigenvalues relate to the actual magnetic transitions from one state to the next?
This has already been adressed here: https://www.physicsforums.com/showthread.php?t=173896 , but I still didn't get the answer.
The Harmonic Oscillator is fully described (according to my favourite QM book) by the HO Hamiltonian, and the commutation relations between the position and momentum...
Homework Statement
Homework Equations
The Attempt at a Solution
I solved part a) correctly, I believe, giving me
ψ = e^{-(√(km)/\hbar)x^{2}}
and a normalization constant A = ((π\hbar)/(km))^{-1/4}
I'm having difficulty with part b. I'm not exactly sure how I create a...
I have only heard about the use of ladder operators in connection with the harmonic oscillator and spin states. However, I would expect them to be useful in other systems as well.
For example, can we find ladder operators for the discrete states of the hydrogen atom, or any other system with...
I have a homework problem which asks me to compute the second and third excited states of the harmonic oscillator. The function we must compute involves taking the ladder operator to the n-power. My question is this: because the ladder operator appears as so, -ip + mwx, and because I am using it...
Homework Statement
I have been given the following problem -
the expectation value of px4 in the ground state of a harmonic oscillator can be expressed as
<px4> = h4/4a4 {integral(-infin to +infin w0*(x) (AAA+A+ + AA+AA+ + A+AAA+) w0 dx}
I think I know how to proceed on other...
I'm am trying to derive the relations:
a|n\rangle=\sqrt{n}|n-1\rangle
a^{\dagger}|n\rangle=\sqrt{n+1}|n+1\rangle
using just the facts that [a,a+]=1 and N|n>=|n> where N=a^{\dagger}a (which implies \langle n|N|n\rangle=n\geq 0). This is what I've done so far:
[a,a^{\dagger}]=1 \Rightarrow...
I'm trying to show that \sum_{m=0}^\infty \frac{1}{m!} (-1)^m {a^{\dagger}}^m a^m =|0 \rangle\left\langle 0|
Where a and {a^{\dagger}} denote the usual annihilation and creation operators. The questions suggests acting both sides with |n> but even if I did that and showed LHS=...=RHS then that...