1

I'm trying to understand the phase-number uncertainty relation for superconductors, \begin{align} \Delta N \Delta \varphi \gtrsim 1. \end{align}

In particular, I'm trying to understand if it holds for the states \begin{align} \vert \psi_\varphi \rangle &= \prod_k (u_k + e^{i\varphi} \nu_k \hat{c}^{\dagger}_{k \uparrow}\hat{c}^{\dagger}_{-k \downarrow}) \vert \psi_0 \rangle \\ \vert \psi_N \rangle &= \int_0^{2\pi} e^{iN\varphi/2} \vert \psi_\varphi \rangle \end{align} of well defined phase and well defined particle number.

It seems that, for the first case $\Delta N$ is finite (estimated by Tinkham to be $\approx 10^9$ - for a macroscopic classic superconductor, I believe), while $\Delta \varphi$ is zero (since $\varphi$ is well defined). For the second case, on the other hand, I would say $\Delta N = 0$ and $\Delta_\varphi = 2 \pi$. In both cases, the product of uncertainties is zero. But if that is the case, the uncertainty relation above would be simply wrong.

Is there a flaw in my train of thought? If not, what is the usefulness of the uncertainty relation above if it does not apply to the two most basic states in this context?

Lucas Baldo
  • 1,558
  • 1
    This 10^9 looks obscure. Where would any scale come in? Also, since $\Delta\varphi<\2pi$, this statement only makes sense as long as $\Delta\varphi\ll 2\pi$. – Norbert Schuch Jul 23 '21 at 22:53
  • The $10^9$ comes from $\Delta N \approx (\langle N \rangle T_c/T_F)^{1/2}$, where $\langle N \rangle \approx 10^{24}$. The scale comes from physical parameters of usual superconductors and Avogadro's number, I suppose. About your second point, which statement are you referring to? – Lucas Baldo Jul 23 '21 at 23:55
  • 1
    Ok, so the 10^9 has nothing to do with the formula above, but with a specific physical situation it refers to. Regarding point 2, the uncertainty relation. – Norbert Schuch Jul 24 '21 at 00:52
  • Ok, but in that specific physical situation, does the uncertainty relation hold for $\vert \psi_\varphi\rangle$? – Lucas Baldo Jul 24 '21 at 01:22
  • What observable are you using to measure phase? Pegg-Barnett? – Quantum Mechanic Jul 24 '21 at 03:41
  • @QuantumMechanic not sure. I'm just following Thinkham's Introduction to Superconductivity 2nd Ed. This discussion is in page 52, chapter 3.3. The author does not mention an observable related to measuring phase and I'm unfamiliar with the possibilities. – Lucas Baldo Jul 24 '21 at 11:13
  • 2
    This is related to the problem of properly defining a self-adjoint phase operator. See for instance https://physics.stackexchange.com/q/338044/36194 – ZeroTheHero Jul 24 '21 at 13:29
  • @ZeroTheHero Thank you, I'll read up on that – Lucas Baldo Jul 24 '21 at 14:16
  • 2
    Additional note: there are other discussions on phase operator throughout the site although I don't recall that they address the exact problem of $\Delta N=0$ that you are referring to. – ZeroTheHero Jul 24 '21 at 14:45

0 Answers0