Have I done this much right? I think I might have made a mistake because I was expecting the contribution to the expectation coming from the kinetic energy to vanish.
Suppose I have a system of N identical bosons interacting via pairwise potential V(\vec{x} - \vec{x}').
I want to show that the expectation of the Hamiltonian in the non-interacting ground state is
\frac{N(N-1)}{2\mathcal{V}}\widetilde{V}(0)
where
\widetilde{V}(q) = \int d^3 \vec{x}...
Do analysis.
I have the same interests as you and I would say that in retrospect taking algebra over analysis was a bad decision. Yes, lie groups and lie algebras play an important role in advanced theoretical physics like particle theory, but you won't be covering that; just finite group...
Any two-level system can be written in the form e^{-i\phi/2}\cos(\theta/2) | 0 \rangle + \sin\theta(\theta/2) e^{i\phi/2}|1\rangle justifying the Bloch sphere interpretation.
The density operator of the two-level system can be expanded in the basis of Pauli matrices...
Hi all,
Can anybody please explain to me the connection between Rabi oscillations and spin-1/2 systems?
I believe the connection lies in the bloch sphere and the ability to represent the spin-1/2 system by a superposition of Pauli matrices but I'm just not getting it.
Thanks
I suppose what I want to show is that the term
\sum_{\vec{k},\vec{k}',\alpha,\alpha'}\int d^3 \vec{x} c^\ast_{\vec{k}'\alpha'}c_{\vec{k}\alpha} (\vec{k}\cdot \vec{u}^\ast_{\vec{k}'\alpha'})(\vec{k'}\cdot \vec{u}_{\vec{k}\alpha})
vanihses. For then,
\frac{1}{2}\int d^3...
I'm trying to get from the magnetic vector potential
\vec{A}(\vec{x},t) = \frac{1}{\sqrt{\mathcal{V}}}\sum_{\vec{k},\alpha=1,2}(c_{\vec{k}\alpha}(t) \vec{u}_{\vec{k}\alpha}(\vec{x}) + c.c.)
where
c_{\vec{k}\alpha}(t) = c_{\vec{k}\alpha}(0) e^{-i\omega_{\vec{k}\alpha}t}...
What material do they cover and what are your interests.
Personally, I majored in physics/mathematics and took algebra over analysis.
The main use of analysis in physics is probably residue calculus which I taught myself when I needed it.
Hi all,
If I have the wave function of a system, then the expectation of position is easily visualized as the centroid of the distribution.
Does anyone know how to visualize the expectation of velocity given just the postion-space wavefunction (real and imaginary parts)
Homework Statement
An +x-polarized electron beam is subjected to magnetic field in the y-direction. What is the probablity of measuring spin +x after a period of time t.
Homework Equations
Time evolution operator U = e^{-i/\hbar \hat{H} t}
The Attempt at a Solution
Since the...
Hi all,
Consider the the number of distinct permutations of a collection of N objects having multiplicities n_1,\ldots,n_k. Call this F.
Now arrange the same collection of objects into k bins, sorted by type. Consider the set of permutations such that the contents of any one bin after...
I'm trying to evaluate the expectation of position and momentum of
\exp\left(\xi (\hat{a}^2 - \hat{a}^\dag^2)/2\right) e^{-|\alpha|^2} \sum_{n=0}^\infty \frac{\alpha^n}{\sqrt{n!}} |n\rangle}
where \hat{a},\hat{a}^\dag are the anihilation/creation operators respectively.
Recall \hat{x}...
Conservation of energy eh? I like that explanation.
What assurance do we have that the photon does not exchange energy with its surroundings in passing from one medium to another?
Is it it possible to `bump up' the energy of a photon that is part of a self-propagating electromagnetic...
This is something I really should know but found I was unable to explain it to myself. When a ray of light passes from one medium to another its frequency remains invariant, but it slows down, forcing the wavelength to decrease according to c = \nu\lambda.
The frequency of the wave will...
Thanks for replying.
How do you define x',p' etc.? You also say that i[H,X] is simply related to i[H',x] which is indeed easy to compute. In fact
i[H',x] = \frac{c^2\vec{p}}{E}
I'm afraid I don't say what the simply relationship is? Could you please expand upon that?
If one takes the derivative of the position operator in the Dirac Hamiltonian, the result is \dot{\vec{x}} = c \vec{\alpha}. This, however, disagrees with the classical limit in which \dot{\vec{x}}\sim \dot{\vec{p}}/m.
I'm trying to show that the time derivative of the position operator...
The equation of motion for an observeable A is given by \dot{A} = \frac{1}{i \hbar} [A,H].
If we change representation, via some unitary transformation \widetilde{A} \mapsto U^\dag A U is the corresponding equation of motion now
\dot{\widetilde{A}} = \frac{1}{i \hbar}...
Given the Hamiltonian H = \vec{\alpha} \cdot \vec{p} c + \beta mc^2,
How should one interpret the commutator [\vec{x}, H] which is supposedly related to the velocity of the Dirac particle? \vec{x} is a 3-vector whereas H is a vector so how do we commute them. Is some sort of tensor product in...
Ahh,
If A\psi belongs to the b-eigenspace then we can express it as
A\psi = a_1\psi_1 + a_2\psi_2
A(\psi_1 + \psi_2) = a_1\psi_1 + a_2\psi_2
so A\psi_i = a_1\psi_i by linear independence.
In the degenerate case, I'm not quite understanding what A\psi being an eignestate of B has to do with being able to diagonalize A in the b-eigenspace?
Could someone please help me understand this?
Since no one has yet replied, let me see if I can get any further withe the first part.
First of all the definition of Lorentz invariance is that the equation remains true in arbitrary intertial reference frames.
The quantity
A_{\mu\lambda} = \sum_{a}...
Yes, I think there is. Note that you have two differential equations: one first order and one second order (the Lagrange equation). Hint: use the second order. But since you are interested in the shape, you need to change from time derivatives to derivatives wrt \phi. Question: what is the...
Hang on. It just occured to me that the implication \vec{k} \to - \vec{k} \implies d\vec{k} \to -d\vec{k} might not be valid since we're doing volume integrals here. Perhaps this saves me?
Hi nrqed,
Thanks for your reply. The page reference is p. 23 Eq. (23).
We want to show that
D(x) = - i\int \frac{d^3\vec{k}}{(2\pi)^32\omega_k}[e^{-i(\omega_k t - \vec{k}\cdot\vec{x})}\theta(x^0) + e^{i(\omega_k - \vec{k}\cdot\vec{x})}\theta(-x^0)].
Although Zee states the contour...
Homework Statement
I'm trying to show that the general form of the propagator is
D(x) = - \int \frac{d^3k}{(2\pi)^32\omega_k}[e^{-i(\omega_k t - \vec{k}\cdot\vec{x})}\theta(x^0) + e^{i(\omega_k - \vec{k}\cdot\vec{x})}\theta(-x^0)]
but my answers always seem to differ by a sign.
Homework...
Sigh. Nevermind, there was a typo in my second integral, Eq. (15) is actually
Z = \int D \varphi e^{i \int d^4 x [-\frac{1}{2}\varphi(\partial^2+m^2)\varphi + J\varphi]} which can be obtained easily by integration by parts on the (\partial \varphi)^2 term:
\int d^4 x\, (\partial \varphi)^2 =...
Homework Statement
I'm studying from Zee's QFT in a nutshell. On page 21, I don't understand how he uses integration by parts to get from Eq (14) to Eq (15), ie from
Z = \int D \varphi e^{i \int d^4 x \{ \frac{1}{2}[(\partial \varphi)^2 - m^2 \varphi^2] + J\varphi \}}
to
Z = \int D \varphi...
I figured it out: For future knowledge, the trick is to use that the matrix A must be symmetric, and thus the derivative of the \frac{1}{2}\mathbf{x}^t A^{-1} \mathbf{x} with respect to x_i, say can be written
\sum_n (A^{-1})_{in} x_n.