Vortex Nucleation Critical velocity using the Uncertainty Principle

[Ted55]

Homework Statement

Calculate the critical velocity for vortex nucleation for HeII passing through a capillary of inner diameter 0.1mm at T=0.

The energy of the ring is E = 1/2 * ρ * k^2 r ( ln(8r/a) -7/4)
You may estimate the rate of vortex nucleation from the uncertainty principle.
Where ρ is the superfluid density, a is the radius of the normal-fluid core of the vortex ring, and k is the circulation quantum.

Relevant Equations

ΔEΔt≥hbar/2

Hi there, I'm very stuck on this problem when approaching it like this. I know I could use the Landau Criterion for rotons but that's not accepted here, it wants the approach to come from the uncertainty principle.

My thinking is along these lines:
There will be a change in chemical potential of the superfluid flowing through the capillary, we can call it Δμ.
Then we can say that we have two energies due to this potential gradient, one at one end of the capillary, and one at the other end, we can say that the energy at the end of the capillary is: E_2 = 1/2 * ρ * k^2 r ( ln(8r/a) -7/4) * Δμ.
The energy at the 'start' of the capillary is simply E_1 = 1/2 * ρ * k^2 r ( ln(8r/a) -7/4), such that we have a change in energy of:

ΔE = E_2 -E_1 = 1/2 * ρ * k^2 r ( ln(8r/a) -7/4)(Δμ -1).

The critical velocity would imply a minima condition and so we would change the inequality above to: ΔEΔt=hbar/2.
I can sub my expression for ΔE in but it doesn't seem to get me anywhere, how do I calculate the vortex nucleation from this potential gradient? I fear I am approaching it all wrong!

Many thanks in advance!

 

[Nate810]

I think you are on the right track. The way to approach this is to solve for the critical velocity, v_c, at which a vortex will nucleate in the superfluid. To do this, we need to use the Heisenberg Uncertainty Principle, which states that the product of the position uncertainty, Δx, and momentum uncertainty, Δp, must be greater than or equal to Planck's Constant, h.Now, the position uncertainty can be related to the velocity of the superfluid. Since the velocity of the superfluid is proportional to the gradient of the chemical potential, Δμ, we can say that Δx = v_c/Δμ. Also, the momentum uncertainty can be related to the energy uncertainty, ΔE, such that Δp = ΔE/v_c. Substituting these two equations into the Heisenberg Uncertainty Principle and rearranging for v_c, we get:v_c = h/(Δx*Δp) = h/(1/Δμ * ΔE/v_c) = h*Δμ/ΔE.Finally, substituting your expression for ΔE into this equation should give you an expression for the critical velocity at which a vortex will nucleate in the superfluid.