1
vote
1answer
13 views

Positivity and Complete positivity of Simon Map

Simon map in a specific basis is defined as $$ \left[ {\begin{array}{ccc} A & B & C \\ D & E & F \\ G & H & I \\ \end{array} } \right] \rightarrow \left[ ...
1
vote
1answer
34 views

How can we show that a map is a completely positive map?

I am doing a homework problem where I have to find whether the map $$ \rho ~\rightarrow~ {\rm tr}(\rho) I - \rho $$ is completely positive. If the map is not completely positive, a counter-example ...
-1
votes
0answers
13 views

Reversing Summation and Product of Sequences [closed]

I am working on a proof for some homework, so I will leave all details out. I can prove it if this step is mathematically sound: ...
1
vote
0answers
59 views

Continuity in a physical context

I'm currently trying to solve an exercise for my quantum mechanics class and have run into a bit of a jam: Suppose we have the following potential : $V(x) = 0$ if $x > |a/2|$ but $V(x) = V_0$ if ...
3
votes
3answers
78 views

Unitary invariance

Why is it that for any non-negative matrix $M$ and unitary matrix $U$, we have $$\sqrt{UMU^\dagger}=U\sqrt{M}U^\dagger$$? This question has to do with Problem 2c from this sheet. I think I am ...
0
votes
0answers
49 views

Quantum Mechanics Angular Momentum Bra-ket notation

Show that if the state $$ \rvert\gamma\rangle $$ Is real then the expectation value of each component of the angular momentum is zero. Does this imply theangular momentum is zero? My Work: $$ ...
4
votes
1answer
77 views

Quantum Hermiticity Bra-Ket notation please

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
1answer
28 views

calculating degenerancy

Given a function of two positive integers $n_x^2+n_y^2$. $n_x^2+n_y^2=50$ has three combinations of $n_x$ and $n_y$ that result in $n_x^2+n_y^2=50$: $$n_x=7,n_y=1$$ $$n_x=5,n_y=5$$ $$n_x=1,n_y=7$$ ...
2
votes
0answers
35 views

Fock Subspaces and Weight Vectors

I've got an assignment due in a few hours, and I'm at a complete loss as to how to even start it, really. I haven't encountered any Dirac notation before, so I'm having a lot of trouble attempting the ...
2
votes
1answer
68 views

Quantum Hamiltonian commuting with the Pauli-Runge vector.

I have to prove that $[A_j, H] = 0$, with; $$\vec{A} = \frac{1}{2Ze^{2}m}(\vec{L} \times \vec{P} - \vec{p} \times \vec{L}) + \frac{\vec{r}}{r}$$ $$H = \frac{p^2}{2m} - \frac{Ze^2}{r}$$ And, $Z, e, ...
0
votes
0answers
22 views

Commutation with a Hamiltonian analogue

I've been given the problem of showing the following commutation; $$[A_{j} , H] = 0$$ With $H = \frac{p^2}{2m} - \frac{Ze^2}{r}$. Now, I'm assuming that the $A_j$ are Runge vectors (but, they might ...
1
vote
1answer
67 views

Calculate spin wave function given probabilities of its alignment along 2 axes

Problem: An $e^{-}$ exists in such a state that the probability of its spin aligning across the $x_{(+)}$ axis is $P_{x+}=1/2$ and across the $y_{(+)}$ axis is $P_{y+}=1/2$ as well. What is the spin ...
0
votes
1answer
55 views

Angular Momentum commuting with Hamiltonian

I've been given an assignment where I have to prove that the angular momentum operators $L_j = \varepsilon_{jkl}q_{k}p_{l}$ commute with the Hamiltonian, given as $H = \frac{p^2}{2m} + V(r)$. Now, I ...
1
vote
0answers
34 views

Statement about density operators, proof feedback

I'm trying to solve the following exercise: Let $V,W$ be finite dimensional inner product spaces over $\mathbb{C}$. Show that for every $\psi \in V\otimes W$ with $\langle \psi, \psi ...
4
votes
5answers
110 views

General solution of $\frac{d^{2}u}{d\rho^{2}}=\frac{l\left(l+1\right)}{\rho^{2}}u$

In his discussion of the radial wave function of hydrogen Griffiths (Introduction to Quantum Mechanics, 2nd ed, p.146) gives the general solution ...
2
votes
2answers
193 views

Instance of Ehrenfest's Theorem

Please Help me to fill in the gaps Show $$ \frac{\text d \langle {p} \rangle}{ \text{d} t} =\left\langle - \frac{ \partial V }{\partial x} \right\rangle .$$ $$\frac{\text d \langle {p} ...
0
votes
1answer
106 views

Partial trace of a system with isolated evolution

Let $\rho_{AB}$ be the state of a composite quantum system with state space $H_A\otimes H_B$ (two finite dimensional Hilbert spaces). Now assume that $A$ and $B$ are isolated and suffer a unitary ...