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Feynman's Theory of Superfluid Helium-4

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Feynman's Theory of Superfluid Helium-4 Superfluid Helium-4 Noah Miller Columbia University March 6, 2018 Noah Miller (Columbia University) Superfluid Helium-4 March 6, 2018 1 / 143 Why is helium a liquid at low temperatures? Table of Contents 1 Why is helium a liquid at low temperatures? 2 Transition to superfluid 3 Multi-particle wave-function and phonons 4 Dispersion curve 5 Critical velocity (first attempt) 6 Irrotational flow 7 Vortex lines 8 Vortex rings 9 Critical velocity (second attempt) 10 References Noah Miller (Columbia University) Superfluid Helium-4 March 6, 2018 2 / 143 Why is helium a liquid at low temperatures? Helium is small inert Noah Miller (Columbia University) Superfluid Helium-4 March 6, 2018 3 / 143 Why is helium a liquid at low temperatures? Helium atoms are neutral, but... Noah Miller (Columbia University) Superfluid Helium-4 March 6, 2018 4 / 143 Why is helium a liquid at low temperatures? they still have weak inter-atomic interactions when you account for the forces between all individual particles Noah Miller (Columbia University) Superfluid Helium-4 March 6, 2018 5 / 143 Why is helium a liquid at low temperatures? This creates the London dispersion force. Using (second order) perturbation theory, with the extra interactions taken as the perturbing Hamiltonian, we get 3 IAIB αAαB EAB = − 6 2 IA + IB R where EAB is the energy between atoms A and B IA and IB are the first ionization energies αA and αB are the dipole polarizabilities R is the inter-atomic distance Noah Miller (Columbia University) Superfluid Helium-4 March 6, 2018 6 / 143 Why is helium a liquid at low temperatures? Helium is very small. Therefore its polarizability is very small. Therefore its inter-atomic attraction is very small. Noah Miller (Columbia University) Superfluid Helium-4 March 6, 2018 7 / 143 Why is helium a liquid at low temperatures? Noah Miller (Columbia University) Superfluid Helium-4 March 6, 2018 8 / 143 Why is helium a liquid at low temperatures? But how can it remain a liquid at absolute zero, when all motion stops? Shouldn't that small attraction still be enough to make it a solid? http://www.columbia.edu/∼vjd1/solid-liquid-gas.htm Noah Miller (Columbia University) Superfluid Helium-4 March 6, 2018 9 / 143 Why is helium a liquid at low temperatures? Answer: the atoms have some momentum from the uncertainty principle! Because helium is so light, this momentum is large enough to keep it a liquid. Noah Miller (Columbia University) Superfluid Helium-4 March 6, 2018 10 / 143 Why is helium a liquid at low temperatures? Lindemann criterion: crystals liquefy when the amplitude of vibration δx is about 5% − 15% the inter-atomic spacing ∆x Noah Miller (Columbia University) Superfluid Helium-4 March 6, 2018 11 / 143 Why is helium a liquid at low temperatures? We can approximate the inter-atomic energy using the Lennard-Jones potential. ! σ 12 σ 6 E = " − 2 r r Noah Miller (Columbia University) Superfluid Helium-4 March 6, 2018 12 / 143 Why is helium a liquid at low temperatures? At equilibrium, the potential is approximately harmonic 1 V ≈ k(δx)2 2 with " k = 36 σ2 Noah Miller (Columbia University) Superfluid Helium-4 March 6, 2018 13 / 143 Why is helium a liquid at low temperatures? From the uncertainty principle ∆x∆p ≈ ~ we can approximate the kinetic energy: p2 2 KE = ≈ ~ 2m 2m(∆x)2 Noah Miller (Columbia University) Superfluid Helium-4 March 6, 2018 14 / 143 Why is helium a liquid at low temperatures? We can find δx via the conservation of energy. " V = KE =) 18 (δx)2 = KE σ2 Assuming σ ≈ ∆x and ignoring constants, we get r δx KE ≈ ∆x " Noah Miller (Columbia University) Superfluid Helium-4 March 6, 2018 15 / 143 Why is helium a liquid at low temperatures? For helium, " = 3:9 × 10−3eV σ = 2:2A˚ ∆x = 3:7A˚≈ σ m = 6:6 × 10−27kg =) KE = 3:8 × 10−5eV r s δx KE 3:8 × 10−5eV ≈ = = 10% ∆x " 3:9 × 10−3eV So it is plausible that helium is a liquid at 0K! values from http://www.cengage.com/resource uploads/downloads/1111988617 420953.pdf Noah Miller (Columbia University) Superfluid Helium-4 March 6, 2018 16 / 143 Transition to superfluid Table of Contents 1 Why is helium a liquid at low temperatures? 2 Transition to superfluid 3 Multi-particle wave-function and phonons 4 Dispersion curve 5 Critical velocity (first attempt) 6 Irrotational flow 7 Vortex lines 8 Vortex rings 9 Critical velocity (second attempt) 10 References Noah Miller (Columbia University) Superfluid Helium-4 March 6, 2018 17 / 143 Transition to superfluid Helium becomes a liquid at 4K. But at 2:17K it becomes a superfluid! Noah Miller (Columbia University) Superfluid Helium-4 March 6, 2018 18 / 143 Transition to superfluid Leggett, p.71 Noah Miller (Columbia University) Superfluid Helium-4 March 6, 2018 19 / 143 Transition to superfluid https://en.wikipedia.org/wiki/Lambda point Noah Miller (Columbia University) Superfluid Helium-4 March 6, 2018 20 / 143 Transition to superfluid Superfluid helium flows without viscosity under a critical velocity of about 20 cm=s. It just keeps going! This produces some strange phenomena. Noah Miller (Columbia University) Superfluid Helium-4 March 6, 2018 21 / 143 Transition to superfluid It can leak through cracks far too small for other liquids to fit through https://www.youtube.com/watch?v=2Z6UJbwxBZI Noah Miller (Columbia University) Superfluid Helium-4 March 6, 2018 22 / 143 Transition to superfluid It can climb up walls via microscopic films https://en.wikipedia.org/wiki/Superfluidity Noah Miller (Columbia University) Superfluid Helium-4 March 6, 2018 23 / 143 Transition to superfluid (The reason for this doesn't seem understood. The common explanation is that, like all liquids, there is a microscopic layer of helium on the walls of the container called the Rollin film, which is attracted to the atoms on the wall. Because helium can flow with no resistance, it can flow right up and over the wall. This is probably incorrect.) Noah Miller (Columbia University) Superfluid Helium-4 March 6, 2018 24 / 143 Transition to superfluid And many other things too. We will give a qualitative first-principles explanation of some properties of superfluid helium. Noah Miller (Columbia University) Superfluid Helium-4 March 6, 2018 25 / 143 Multi-particle wave-function and phonons Table of Contents 1 Why is helium a liquid at low temperatures? 2 Transition to superfluid 3 Multi-particle wave-function and phonons 4 Dispersion curve 5 Critical velocity (first attempt) 6 Irrotational flow 7 Vortex lines 8 Vortex rings 9 Critical velocity (second attempt) 10 References Noah Miller (Columbia University) Superfluid Helium-4 March 6, 2018 26 / 143 Multi-particle wave-function and phonons A helium-4 atom contains two protons two neutrons two electrons It is a spin 0 boson. (Helium-3 is a fermion but we will not discuss it.) Noah Miller (Columbia University) Superfluid Helium-4 March 6, 2018 27 / 143 Multi-particle wave-function and phonons The quantum state of a collection of atoms is a multi-particle wave-function. Ψ(~x1; ~x2; ~x3;:::; ~xN ) Each x~i is the position of atom i. Noah Miller (Columbia University) Superfluid Helium-4 March 6, 2018 28 / 143 Multi-particle wave-function and phonons As the particles are bosons, it must be completely symmetric under any permutation of atoms. Ψ(~x3; ~x69; ~x420;:::; ~x1) = Ψ(~x1; ~x2; ~x3;:::; ~xN ) Noah Miller (Columbia University) Superfluid Helium-4 March 6, 2018 29 / 143 Multi-particle wave-function and phonons So a wave-function Ψ assigns a complex number to each possible configuration. The bigger jΨj2 is for a configuration, the more likely it is. Noah Miller (Columbia University) Superfluid Helium-4 March 6, 2018 30 / 143 Multi-particle wave-function and phonons Let us call the ground state wave-function Ψ0. What is Ψ0 like qualitatively? Noah Miller (Columbia University) Superfluid Helium-4 March 6, 2018 31 / 143 Multi-particle wave-function and phonons N X 2 1 X H^ = − ~ r2 + V (jx~ − x~ j) 2m i 2 i j i=1 i6=j Noah Miller (Columbia University) Superfluid Helium-4 March 6, 2018 32 / 143 Multi-particle wave-function and phonons Basic facts about definite energy states: jΨj2 is large when potential energy of configuration is small Ψ has higher energy when the gradient is large Basic facts about ground states: Completely real (up to global phase) Never zero, no \nodes" Noah Miller (Columbia University) Superfluid Helium-4 March 6, 2018 33 / 143 Multi-particle wave-function and phonons So we take Ψ0 to be real and positive everywhere. It will be very small when atoms overlap, and gradually grow as they get further apart. Noah Miller (Columbia University) Superfluid Helium-4 March 6, 2018 34 / 143 Multi-particle wave-function and phonons Value of Ψ0 per configuration: Ψ0 large Ψ0 small Ψ0 minuscule Noah Miller (Columbia University) Superfluid Helium-4 March 6, 2018 35 / 143 Multi-particle wave-function and phonons Ψ0 is large when the atoms are evenly spaced out, and small when there are density variations and clumps. Noah Miller (Columbia University) Superfluid Helium-4 March 6, 2018 36 / 143 Multi-particle wave-function and phonons That is the case for the lowest energy state Ψ0. What about the states with slightly higher energy? Noah Miller (Columbia University) Superfluid Helium-4 March 6, 2018 37 / 143 Multi-particle wave-function and phonons The fluid will have a small energy and the multi-particle wave-function will have a few \nodes." (Energy eigenstates can still be taken to be real.) Call our state ΨE . Noah Miller (Columbia University) Superfluid Helium-4 March 6, 2018 38 / 143 Multi-particle wave-function and phonons Because we want ΨE to have a small energy, we want to minimize its gradient as much as possible.
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