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  1. B

    Bloch sphere and mixed stats

    Homework Statement What is reduced density matrix ##\rho_A## and the Bloch vector representation for a state that is 50% ##|0 \rangle## and 50% ##\frac{1}{\sqrt{2}}(|0 \rangle + |1 \rangle)## Homework Equations The Attempt at a Solution [/B] I haven't seen many (any?) examples of this so...
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    Finite Temperature matrix

    I got it figured it out. I just couldn't find it in my textbook. Turns out in was in recommended reading I guess. Thanks for the help though.
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    Finite Temperature matrix

    Homework Statement Consider the Hamiltonian ##H=\begin{bmatrix} 0& \frac{-iw}{2}\\ \frac{iw}{2} & 0 \end{bmatrix}## Write the finite temperature density of the matrix ##\rho(T)## Homework Equations ##\beta=\frac{1}{kT}## The Attempt at a Solution The initial part of the problem had me find...
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    Double Well Potential

    It's no problem. So, I guess I have the 4 equations that I should need, I'm just not sure about their relationships at the boundary conditions.
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    Double Well Potential

    I don't understand quite what mean by ##x=L##. L isn't a boundary but a coefficient. Do I need to treat this as two different double wells since I don't know the energy? I think I understand the equation the other student had. It is describing the function on the right side of the well. And...
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    Double Well Potential

    I guess I'm stuck at this then. Well, I know that A must be zero so the function that describes the range where ## 0\leq x < \frac{\pi}{2}-\frac{a}{2} ## and ##\frac{\pi}{2}+\frac{a}{2} <x\leq \pi## is##\psi_1(x)=Bcos(\sqrt{E}x)## For ##\psi_2## and ##\psi_3## I can't apply the same...
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    Double Well Potential

    Homework Statement Consider the Hamiltonian ##H=-\frac{d^2}{dx^2}+V(x)## ##V(x) =\begin{cases} \infty & x < 0 ,x>\pi \\V_0 & \frac{\pi}{2}-\frac{a}{2} \geq x \leq \frac{\pi}{2}+\frac{a}{2} \\ 0 &elsewhere\end{cases}## Use the boundary conditions at x=0 and x=##\pi## to set two of the...
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    Electron position at time t

    Homework Statement Consider the electron on a six site chain. The Hamiltonian is: ## H = \begin{pmatrix} 0 & -1 & 0 & 0 & 0& 0\\ -1 & 0 & -1 & 0 & 0& 0\\ 0 & -1 & 0 & -1 & 0& 0\\ 0 & 0 & -1 & 0 & -1& 0\\ 0 & 0 & 0 & -1 & 0& -1\\ 0 & 0 & 0 & 0 & -1& 0\\...
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    Compute e^(itH)

    So, then I would just need to sum ##\frac{(1t)^n}{n!}## and ##\frac{(2t)^n}{n!}## from 0 to infinity. giving me a diagonal matrix of just I incorrectly wrote the exponential before, it should have been ##e^{-iHt}##. Forgot the negative. ##H_{diagonal} = \begin{pmatrix} e^{-it} & 0 & 0...
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    Compute e^(itH)

    This is for a physics problem but I suppose I could have put this part in math as well. It is Hamiltonian for an electron on a six site chain. I've gotten eigenvalues from using online calculators, actually doing all of this by hand seems like a huge time sink. I don't get exact values from...
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    Compute e^(itH)

    Homework Statement Compute, using wolfram is an option, eitH where H is: ##H = \begin{pmatrix} 0 & -1 & 0 & 0 & 0& 0\\ -1 & 0 & -1 & 0 & 0& 0\\ 0 & -1 & 0 & -1 & 0& 0\\ 0 & 0 & -1 & 0 & -1& 0\\ 0 & 0 & 0 & -1 & 0& -1\\ 0 & 0 & 0 & 0 & -1& 0\\ \end{pmatrix}## Homework...
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    Time evolution, Hamiltonian

    Okay, it all makes sense to me now. I'll work on the rest of this. I'll probably just make a new post if I need additional help since it won't likely be until tomorrow Thank you.
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    Time evolution, Hamiltonian

    if ##H=\left|i\right>\left<i\right|+\left|j\right>\left<j\right|## ##H= \begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix} ## and if ##H=\left|i\right>\left<j\right|+\left|j\right>\left<i\right|## ##H= \begin{pmatrix} 0 & 1\\ 1 & 0 \end{pmatrix} ## I think that if I had a...
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    Time evolution, Hamiltonian

    I think I fixed it. All of the copying and pasting latex messed me up.
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    Time evolution, Hamiltonian

    Okay, so something like: ##H= \begin{pmatrix} 0 & \frac{w}{2}\\ \frac{w}{2} & 0 \end{pmatrix} =\frac{w}{2}[(0)(|u_z\rangle \langle u_z|+(|u_z\rangle \langle u_z|)+(|d_z\rangle \langle d_z|+(|d_z\rangle\langle d_z|)(0)]## ##=\frac{w}{2}[|u_z\rangle \langle u_z|+|d_z\rangle \langle...
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    Time evolution, Hamiltonian

    Not explicitly. They only ever gave a matrix example as above and then "multiplied" it against a state and gave the result. So, as far as I can confidently say, I only know what an operator multiplied against a state looks like. The example I got was: ##H= \begin{pmatrix} \frac{w}{2} &...
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    Time evolution, Hamiltonian

    Homework Statement A magnetic field pointing in ##\hat{x}##. The Hamiltonian for this is: ##H= \frac{eB}{mc}\begin{pmatrix} 0 & \frac{1}{2}\\ \frac{1}{2} & 0 \end{pmatrix}## where the columns and rows represent ##{|u_z\rangle, |d_z\rangle}##. (a) Write this out in Dirac...
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    Gaussian Wave packet

    Okay, so I got a high number from the integral 376.99 and found that the 2 cosine functions from my ##|\psi(x)|^2## basically added nothing (both came out to ~##10^{-15}##). So, I have ##N(376.99)=1##
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    Gaussian Wave packet

    I'm just trying to figure out the order/process of things. I have done the integral of ##\psi## and it's complex conjugate. So, I guess when I'm trying to do the integral to find ##\frac{1}{N}## am I suppose to do the initial integral with those limits then square it or take the integral of...
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    Gaussian Wave packet

    I keep forgetting about N. In order to find it do I take the initial integral and just use the ##\pm \frac{2}{0.1} \pi## limits? I plugged it into wolfram and it gave me something crazy back.
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    Gaussian Wave packet

    Yeah, he did a really poor job of explaining clearly what he wanted. One of his habits so far. He originally had the first question as simply find N with bounds neg infinity to infinity. But changed it today when it was clear that this wasn't possible. So I got ## | \psi(x)|^2## now but the...
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    Gaussian Wave packet

    Homework Statement Wave function of an electron: ##\frac{1}{N}∫_a^b[e^{ik_0x}(1+\frac{1}{2}e^{i\frac{0.1}{2}x}+\frac{1}{2}e^{-i\frac{0.1}{2}x})]dx## The integrand becomes zero both to the left and right of x = 0 . Let a be the first time it hits zero to the left and b the first time it...
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    Normalization of states

    okay, thanks. I think I understand now. I appreciate the help.
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    Normalization of states

    So for the spin question, the bases do matter then? I just need to switch to a common base?
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    Normalization of states

    I guess I just assumed these were common. ##\langle u_z|u_x\rangle## ##|u_x\rangle=\frac{1}{√2} |u_z\rangle + \frac{1}{√2} |d_z\rangle## ##\frac{1}{√2}(\langle u_z|u_z\rangle) + \frac{1}{√2}((\langle u_z|d_z\rangle)## ##\frac{1}{√2}(1) + \frac{1}{√2}(0)## So ##\langle u_z|u_x\rangle =...
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    Normalization of states

    Homework Statement These are rather simple questions but the rules for all of this are not quite clear to me yet. I'm to determine whether or not the following states are "legal" and if not I should normalize them. a. ##\frac{1}{√385} ∑_{x=1}^{10}x^2 |x>## b. ##\frac{1}{√2}...
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    Field on a plane from line charge

    ##\rho## is the charge density ##\lambda## I was trying to use ##\rho## in the general sense but I sort of got ahead of myself putting it in there. ##E(r)=k\int\frac{1}{r^2}dq## ##dq=\lambda dz## so, ##E(r)=k\int\frac{\lambda}{r^2}dz## As soon as I walked away from the computer I...
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    Field on a plane from line charge

    Homework Statement A wire at extends for -L to L on the z-axis with charge ##\lambda##. Find the field at points on the xy-plane Homework Equations ##E(r)=k\int\frac{\rho}{r^2}dq## ##k=\frac{1}{4\pi ε_o}## The Attempt at a Solution First time I've looked for field on a plane so I...
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    Comoving redshift and area of sphere

    Homework Statement Standard candles may be used to measure the "luminosity distance", using DL = (L/4F)1/2, where L is the source's intrinsic luminosity, and F is the observed flux. Inthis problem you will relate the luminosity distance to the previously discussed angular diameter...
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    Causal contact from opposite sides of Universe?

    Another point I have trouble with is if these 2 points are at the very edge of my visible universe then wouldn't that put them outside of each other's visible universe? They should not be able to see each other at all. Still, my basic question remains as to whether they every had the ability to...
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