I am studying for a comprehensive exam on dynamic contrast enhanced MRI (DCE-MRI) and questions have come up in my practice talks about dynamic susceptibility contrast MRI (DSC-MRI).
In DCE-MRI changes in signal in an artery and the tissue are converted to contrast agent concentration vs time...
When I try to find the eigenvalues of H I get the expression:
(-ε0σz - (qBxħ/4mc)(σ+ + σ-))|+> = ε|+>
and I don't know how to evaluate it properly. I tried plugging these in: σ+ = |+><-|, σ- = |-><+| but I have no idea what to do after that. Are you supposed to expand it so that you're...
Consider a spin system with noninteracting spin 1/2 particles. The magnetic moment of the system is written as:
μ = (ħq/2mc)σ
Where σ = (σx, σy, σz) is the Pauli spin operator of the particle. A magnetic field of strength Bz is applied along the z direction and a second...
Okay so you are saying that <a†a> = αα* is correct, then using the commutation relation:
[a,a†] = aa† - a†a = 1, therefore: aa† = 1 + a†a
So then the expectation value of aa† is: <aa†> = 1 + <a†a> = 1 + αα*
Is that correct?
If that is correct then the expectation values are as follows; <a†>=α*, <a> = α, <aa> = α2, <a†a†> = α*2, and <aa†> = <a†a> = αα*.
This means that the expectation value <x> is: <x> = (ħ/2mω)0.5(<a†> + <a>) = (ħ/2mω)0.5(α* + α)
So the expression (x-<x>)2 = x2 - 2x(ħ/2mω)0.5(α* + α) + (ħ/2mω)(α*...
Consider a particle with mass m oscillates in a simple harmonic potential with frequency ω. The position, x, and momentum operator, p, of the particle can be expressed in terms of the annihilation and creation operator (a and a† respectively):
x = (ħ/2mω)^0.5 * (a† + a)
Thank you, I've been able to get through all of part a. I'm on part b now, and I'm not sure how to express <Ψ|U|Ψ>...
Given that |Ψ> = e-α∑|Φn>
Can <Ψ|U|Ψ> be expressed as:
<Ψ|U|Ψ> = e-α∑<Φn|U|Φn>
So then the expectation value of U in state Ψ, is the infinite sum of the expectation value...
I've tried expressing U in terms of powers of H and got the following:
e(iHt/ħ) = [e(it/ħ)]H = ∑(H^n)/n! = 1 + H + H2/2 + H3/6 +...
What exactly am I supposed to do with this now?... I think my main source of confusion is that I'm unfamiliar with Dirac notation (which unfortunately seems to...
The Hamiltonian of an electron in solids is given by H. We know that H is an Hermitian operator, it satisfies the following eigenvalue equation:
H|Φn> = εn|Φn>
Let us define the following operators in terms of H as:
U = e^[(iHt)/ħ] , S = sin[(Ht)/ħ] , G = (ε -...
Thanks to both of you for the help. I've done as you suggested and solved the commutator and found that it was equal to dA(x)/dx, now the answer I have is:
[px,A(x)] = -iħ*(dA(x)/dx)
Another source I've seen online seems to suggest that the first term should be ħ/i (which is equal to my...
Consider A(x) is an arbitrary function of x, and px is the momentum operator. Show that they satisfy the following condition:
[px,A(x)] = (-i/ħ)*d/dx(A(x))
where [px,A(x)] = pxA(x) - A(x)px
ħ = h/2π
px = (-iħ)d/dx
The Attempt at a Solution
Someone please help me. I have no idea what I'm doing and need to get this assignment done ASAP. I know now that I need to find H first using Ampere's law but am having a hard time understanding how to define my Amperian loop
We have an infinite slab of conducting material, parallel to the xy plane, between z = −a and z = +a, with magnetic susceptibility χm. It carries a free current with volume current density J = J0z/a in the x direction (positive for z > 0, negative for z < 0). The integrated...