1
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2answers
97 views

Determining the Wave Function From Initial Conditions

This is Problem 2.6 (b) in Griffiths, Intro to QM: A particle in an infinite square well has its initial wave function an even mixture of the first two stationary states: $\Psi(x,0) = ...
1
vote
1answer
79 views

Expectation value of an operator

Suppose we have: $$ \hat{Q}|\psi_1\rangle=q_1|\psi_1\rangle \\ \hat{Q}|\psi_2\rangle=q_2|\psi_2\rangle $$ with $q_1 \neq q_2$. Then consider the state: $$ ...
1
vote
1answer
52 views

Atom state vectors kets

An atom with two energy levels has 2 states (excited and ground), represented by kets $|e\rangle$ and $|g\rangle$ respectively. The atom has energy $\frac{1}{2}E_\theta$ when excited and ...
1
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0answers
61 views

How to make a base tranformation for a linear operator in QM? [closed]

I have 2 bases A and B with the following kets: Base A: $|a_1\rangle$ and $|a_2\rangle$ Base B: $|b_1\rangle = \frac{1}{\sqrt2} \cdot(|a_1\rangle + i\cdot|a_2\rangle)$ $|b_2\rangle = ...
1
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0answers
50 views

Quantum Rigid Rotor Perturbation

As the title says, I have a rigid rotor with a perturbation given below $$H=\frac{L^2}{2I}-\alpha B L_z.$$ So I know that the eigenvalues of $H$ will be $\ell(\ell+1)/2I -\alpha B m$ where $m$ is our ...
1
vote
1answer
97 views

Normalisation of Linear Harmonic Oscillator - Ladder Operator Method

I was watching the following video on the harmonic oscillator using ladder operators : http://youtu.be/gRdCV9p8sAU?t=30m9s Clicking on the video above will take you to the exact point where my ...
1
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2answers
102 views

Quantum Mechanics: Momentum operator questions [closed]

I'm asked to determine $\hat{P}|\Psi_0\rangle$, $\langle{\hat{P}}\rangle$, and $\langle\hat{P}^2\rangle$ for $$\Psi_0(u) = \psi_0 + 2\psi_1$$ I understand how to make the matrix for $P$ in regards ...
4
votes
1answer
98 views

Orthogonality of summed wave functions

Problem. I know that the two wave functions $\Psi_1$ and $\Psi_2$ are all normalized and orthogonal. I now want to prove that this implies that $\Psi_3=\Psi_1+\Psi_2$ is orthogonal to ...
1
vote
0answers
48 views

How do I properly express adding perturbed states to unperturbed states?

I have a problem set due tomorrow, and the last problem is driving me nuts. Been combing through griffiths trying to find similar examples to no avail, so it'd be greatly appreciated if stackexchange ...
3
votes
1answer
207 views

Example of application of creation/annihilation operators in matrix form

I was wondering how it would sound like the creation/annihilation of particles that we usually do in the context of Dirac formalism, with matrices and vectors. As a reminder we know that: ...
4
votes
1answer
160 views

Continuity domain for momentum operator

I know this is essentially a mathematic question, but I received no answer on math SE. Moreover it has a direct application in physics, so I thought to ask this here too. The momentum operator in one ...
1
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2answers
364 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 ...
2
votes
2answers
1k views

Proving that the hermitian conjugate of the product of two operators is the product of the two hermitian congugate operators in opposite order

I have reach a step in a problem of my quantum mechanics textbook that requires me to prove the following. $$\hat{A}=(\hat{Q}\hat{R})^{\dagger} = \hat{R}^{\dagger}\hat{Q}^{\dagger}$$ I tried to ...
2
votes
2answers
1k 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 ...
1
vote
1answer
509 views

Writing a Wavefunction as a Linear Combination of Eigenstates

We have the following wavefunction for the hydrogen atom: $$\psi(r,\theta,\phi)=\frac{1}{\sqrt{4\pi}}\frac{1}{(2a)^{3/2}}\frac{r}{a}e^{-r/2a}\sin(\theta)\sin(\phi)$$ where $a$ is the Bohr radius. ...
1
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2answers
260 views

Quantum Mechanics Operator Hermiticity

If $A$ and $B$ are Hermitian operators, show that $$C~:=~i[A,B]$$ is Hermitian too. My work: $$\begin{gather} C=i(AB-BA) \\ \langle\psi\rvert C\lvert\phi\rangle = i\langle\psi\rvert ...
2
votes
2answers
135 views

Scalar QFT Fock Space

I want to demostrate the following relation of the normal ordered product: $\Omega\equiv:\exp{\left(-\int d^3k~a^{\dagger}(k)a(k)\right)}:=|0\rangle\langle0|.$ I proved the commutation relation ...
1
vote
0answers
130 views

A basic question about Heisenberg State Kets (in particular the simple harmonic oscillator)

I know base kets in the Heisenberg picture are $U^\dagger |{a}\rangle$ but if the base kets are the base of the hamiltonian, and the hamiltonian is independent of time, are all of the base kets ...
1
vote
2answers
325 views

Wave function decomposition

Problem: Given the wave function $\Psi_0=A\sin^2(\theta)$ along with the Hamiltonian operator of a physical system: $H=\frac{L^2}{2I}+g B L_z$, find the eigenvalues and eigenfunctions of $\hat{H}$ ...
1
vote
3answers
273 views

$\mathrm{Tr}(XY) = \mathrm{Tr}(YX)$?

I'm trying to understand the Dirac notation to understand quantum mechanics better. I'm trying to show the above relation using the Dirac notation. Given $$\mathrm{Tr}(X)~=~\sum_j\langle ...
3
votes
2answers
169 views

Space translation of operators, states, and particle densities

In Sidney Coleman's Lectures he talked about space translations such that $$\tag{1} e^{ia P}\rho(x) e^{-ia P} ~=~ \rho(x-a),$$ but when I expanded the exponentials and used the commutation relation ...
3
votes
2answers
160 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, ...
3
votes
1answer
245 views

Show that for QM operator A: $\int_{-\infty}^{\infty}\psi A^{\dagger}A\psi dx = \int_{-\infty}^{\infty}(A\psi)^*(A\psi)dx $

I need to show for $$A = \frac{d}{dx} + \tanh x, \qquad A^{\dagger} = - \frac{d}{dx} + \tanh x,$$ that $$\int_{-\infty}^{\infty}\psi^* A^{\dagger}A\psi dx = ...
0
votes
3answers
137 views

What is this state as a matrix?

In QM I have the state $\lvert 00 \rangle \langle 00 \rvert$. Can anyone tell me what this would look like as a matrix? I know that $$ \lvert 00 \rangle = \begin{pmatrix} 1 & 1 \\ 0 & 0 ...
0
votes
1answer
685 views

Proving Operator identities (Quantum Physics)

How would I go about showing: $$\hat{A}^{\dagger} + \hat{B}^{\dagger} = \left( \hat{A} + \hat{B} \right) ^{\dagger}$$