What is Ladder operator: Definition and 22 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.
I'm trying to apply an operator to a massless and minimally coupled squeezed state. I have defined my state as $$\phi=\sum_k\left(a_kf_k+a^\dagger_kf^*_k\right)$$, where the ak operators are ladder operators and fk is the mode function $$f_k=\frac{1}{\sqrt{2L^3\omega}}e^{ik_\mu x^\mu}$$...
I'm just trying to follow the below
And I understand all, I think, except what's happened to the term when A hits 1: [A,1] ?
If I'm correct basically we're just hitting on the first operator so reducing the power by one each time of the operator in the right hand bracket
thanks
As far as I know we can express the position and momentum operators in terms of ladder operators in the following way
$${\begin{aligned}{ {x}}&={\sqrt {{\frac {\hbar }{2}}{\frac {1}{m\omega }}}}(a^{\dagger }+a)\\{{p}}&=i{\sqrt {{\frac {\hbar }{2}}m\omega }}(a^{\dagger }-a)~.\end{aligned}}.$$...
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
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
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The textbook states ##|N_{\pm}|^2=\langle jm | J_{\pm}^\dagger J_{\pm} | jm \rangle##...
We learned that we can use the ladder operator to obtain the states of a quantum oscillator. However, I see no direct evidence to show that the solutions are complete. I mean, how can we know the energy state follows E is (E+hw). Why can't we have some more states in between? Does the derivation...
Hi, quick question with A being the lowering operator and A† the raising operator for a QHO
(A A† - 1 + 1/2) ħω [Aψ] = A (A† A - 1 + 1/2) ħω ψ
By taking out a factor of A. Why has the ordering of A A† swapped around? I would have thought taking out a factor of A would leave it as
A (A† - 1 +...
Hey there!
1. Homework Statement
I've been given the operators
a=\sqrt\frac{mw}{2\hbar}x+i\frac{p}{\sqrt{2m\hbar w}} and a^\dagger=\sqrt\frac{mw}{2\hbar}x-i\frac{p}{\sqrt{2m\hbar w}} without the constants and definition of the momentum operator:
a=x+\partial_x and a^\dagger=x-\partial_x with...
In the Griffiths textbook for Quantum Mechanics, It just gives the ladder operator to be
L±≡Lx±iLy
With reference to it being similar to QHO ladder operator. The book shows how that ladder operator is obtained, but it doesn't show how angular momentum operator is derived.
Ive searched the...
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
Define n=(x + iy)/(2)½L and ñ=(x - iy)/(2)½L.
Also, ∂n = L(∂x - i ∂y)/(2)½ and ∂ñ = L(∂x + i ∂y)/(2)½.
with ∂n=∂/∂n, ∂x=∂/∂x, ∂y=∂/∂y, and L being the magnetic length.
Show that a=(1/2)ñ+∂n and a†=(1/2)n -∂ñ
a and a† are the lowering and raising operators of quantum...
Homework Statement
let A be a lowering operator.
Homework Equations
Show that A is a derivative respects to raising operator, A†,
A=d/dA†
The Attempt at a Solution
I start by defining a function in term of A†, which is f(A†) and solve it using [A , f(A†)] but i get stuck after that. Can...
In any textbooks I have seen, vacuum states are defined as:
a |0>= 0
What is the difference between |0> and 0?
Again, what happens when a+ act on |0> and 0?
and Number Operator a+a act on |0> and 0?
If the ladder operator ##a=\sqrt {\frac{m\omega}{2\hbar}}x+\frac{ip}{\sqrt{2m\hbar \omega}}## and ##a^\dagger=\sqrt {\frac{m\omega}{2\hbar}}x-\frac{ip}{\sqrt{2m\hbar \omega}}## then I get that the number operator N, defined as ##a^\dagger a## is worth ##\frac{m \omega...
Homework Statement
What is the effect of the sequence of ladder operators acting on the ground eigenfunction \psi_0
Homework Equations
\hat{A}^\dagger\hat{A}\hat{A}\hat{A}^\dagger\psi_0The Attempt at a Solution
I'm not sure if I'm right but wouldn't this sequence of opperators on the ground...
I'd like to know how to do it without solving via Hermite polynomials, so that I can check by both methods when I solve other problems. I have tried to figure out myself but I need some help.
Let's say H:=x^2+p^2 , and a:= x-ip .
So that [a,a^+]=2\hbar , H a \varphi_i = (E_i -...
Hi!
I don't know much about QM. I'm reading lecture notes at the moment. Angular momentum is discussed. The ladder operators for the angular-momentum z-component are defined, it is shown that <L_z>^2 <= <L^2>, so the z component of angular momentum is bounded by the absolute value of angular...
A Quantum I problem set asks me to graph the first 15 states of the simple harmonic oscillator. Our department uses mathcad heavily, so I think I should write a function that applies the ladder operator repeatedly to generate the wave function. I'm having trouble getting it to actually return a...