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...
A square wire loop of size 2a by 2a lies in the x- y plane with its center at the origin and sides parallel to the x and y axes. A counterclockwise current I runs around the loop.
(a) Find the magnetic field on the z axis. [Answer: Bz = 2μ0Ia2/[(a2 + z2)(2a2 + z2)0.5]]
I don't know, that's what I'm trying to understand. I was trying to express that statement in mathematical form but couldn't figure it out, so I settled on the expression that I have simply because integrating it over the cross section of the wire results in the total current I
A long, straight wire, radius R, carries total current I. The current is distributed in the wire so that the current
density is proportional to s, the distance from the center of the wire.
(a) Write an expression for the current density J in the wire, as a function of s...
If you mean the surface bound charge on the outer surface when you say I haven't accounted for all the charge yet, then I already found that. Thanks for pointing out the sign of the volume bound charge, I found the spot where I missed a minus sign. Now it seems that the total volume bound...
I need to calculate the electric field on the midplane of a uniformly polarized cylinder at a large distance from the center of the cylinder. The question also says that because the distance is large compared to the radius the dipole dominates the multipole expansion...
2 concentric conducting spherical shells, with radii a and 2a, have charge +Q and -Q respectively. The space between the shells is filled with a linear dielectric with permittivity ε(r) = (ε0*a)/(1.5*a - 0.5*r), which varies with distance r.
a) Use Gauss's law to determine...
Sorry probably should have been more specific. The 2 vectors are actually an electric field and electric displacement. Both have only an r component. Based on what you have said I assume that if they are parallel you can just multiply the components together. Since both the electric field...
I'm doing a question that requires me to take the dot product of 2 vectors in spherical coordinates. Both vectors have only an r component, can I just multiply the r components?
The Attempt at a Solution
Sorry i must not have made the problem clear enough... I am looking for the potential in the center of a uniformly polarized cylinder. Since the polarization causes surface charges to only exist on the top and bottom surfaces, I was approximating the situation by looking for the potential...
I used the equation for finding potential due to a surface charge:
V = (1/4*pi*ε) ∫ (σ/r2) da
Where ε is the vacuum permittivity constant, σ is the surface charge density, and r is the distance between the disc and the field point.
Evaluating this expression for the top disc I get:
This is the approach that I used:
I wanted to find an expression for the potential due to the top and bottom discs separately, then sum the 2 expressions to find the total potential, once I have an equation for the total potential I would take the gradient of it to find the electric field...