Unbound States. 6.3 Quantum Tunneling Examples Alpha Decay The Tunnel Diode SQUIDS Field Emission The Scanning Tunneling Microscope

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May Williams
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1 Unbound States 6.3 Quantum Tunneling Examples Alpha Decay The Tunnel Diode SQUIDS Field Emission The Scanning Tunneling Microscope 6.4 Particle-Wave Propagation Phase and Group Velocities Particle-like wave packet Electromagnetic Pulse

The Schrodinger equation d 2!(x) dx 2 =! 2m(E!U 0 )! 2!(x) Barrier Penentration in General has a solution for a barrier potential of the form: Incident and Reflected! x<0 (x) = Ae +ikx + Be!

2 The Schrodinger equation d 2!(x) dx 2 =! 2m(E!U 0 )! 2!(x) Barrier Penentration in General has a solution for a barrier potential of the form: Incident and Reflected! x<0 (x) = Ae +ikx + Be!ikx Classically Forbidden! 0<x<L (x) = Ce +"x + De!"x k = Where: Transmitted! x>l (x) = Fe +ikx 2m(E! 0)! 2 ;! = 2m(U 0! E)! 2 ; sinh 2 [ 2m(U R = 0! E) " L /!] sinh 2 [ 2m(U 0! E) " L /!]+ 4(E /U 0 )(1! E /U 0 ) 4(E /U T = 0 )(1! E /U 0 ) sinh 2 [ 2m(U 0! E) " L /!]+ 4(E /U 0 )(1! E /U 0 ) R 2 +T 2 =1

Outside of the parent nucleus the alpha-particle experiences only the electrostatic repulsion of its two protons with the remaining protons of the

3 Alpha Decay Alpha Decay is a form of radioactivity. Radioactive decay of a nucleus through emission of an alpha-particle. Alpha-particles are helium nuclei. Nucleons are bound inside of the nucleus due to the strong nuclear force. Outside of the parent nucleus the alpha-particle experiences only the electrostatic repulsion of its two protons with the remaining protons of the parent nucleus. U elec = q 1 q 2 4!" 0 r = (2!1.6!10 "19 C)(90!1.6!10 "19 C) 4! (8.85!10 "12 C 2 / N m 2 )(7.4!10 "15 m) U elec = 5.6!10 "12 J = 35MeV But observed energy of alpha particle from this decay is only 4.3 MeV Tunneling!!!!

87x10 "13 J) r 1 = 6.03!10 "14 m = 60.3 fm r 0 = 7.4 fm U elec = q 1q 2 4!" 0 r = (2!1.6!10"19 C)(90!1.6!10 "19 C) 4! (8.85!10 "12 C 2 / N m 2 )(r) U elec (r 0 ) = 5.6!10 "12 J = 35MeV U elec (r 1 ) = 6.

4 Energy of Ejected alpha-particle is same as alpha-particle inside parent nucleus Alpha Decay E = mc 2 = (0.0046u!1.66x10 "27 kg / u)(3!10 8 m / s) 2 E = 6.87!10 "13 J = 4.30MeV Calculation of Transmission Probability Depends on barrier height and width r 1 = q q 1 2 (2!1.6!10 "19 C)(90!1.6!10 "19 C) = 4!" 0 E # 4! (8.85!10 "12 C 2 / N m 2 )(6.87x10 "13 J) r 1 = 6.03!10 "14 m = 60.3 fm r 0 = 7.4 fm U elec = q 1q 2 4!" 0 r = (2!1.6!10"19 C)(90!1.6!10 "19 C) 4! (8.85!10 "12 C 2 / N m 2 )(r) U elec (r 0 ) = 5.6!10 "12 J = 35MeV U elec (r 1 ) = 6.87!10 "13 J = 4.3MeV WKB approximation solution for Transmission through barrier T = exp[!2 r 1 " (2m E!U elec (r) )dr! Decay Rate r 0 rate = v 2r 0 T = 2m! E! 2r 0 T

Mean time to decay = τ=1/rate Alpha Decay Half-Life vs Energy of Alpha Particle Half life t 1/2 =ln(2) τ=0.693 τ Half-Life is very sensitive to alpha particle kinetic energy!

5 Mean time to decay = τ=1/rate Alpha Decay Half-Life vs Energy of Alpha Particle Half life t 1/2 =ln(2) τ=0.693 τ Half-Life is very sensitive to alpha particle kinetic energy!!! α emitting nucleus Energy Mean time to decay Po MeV 4.4 x 10-7 sec Rn Mev 79 seconds Ra MeV 5.3 days Ra MeV 2300 years U MeV 6.5x10 9 years

Tunnel Diode Highly doped E F >E C & E F <E v With no applied voltage no net current As applied voltage slightly increased above zero net current flow increases When number of empty states on the

6 Tunnel Diode Highly doped E F >E C & E F <E v With no applied voltage no net current As applied voltage slightly increased above zero net current flow increases When number of empty states on the p-side decreases the current decreases with applied voltage resulting in a negative resistance region At higher applied voltages diffusion causes the current to increase with voltage again. No time delay for tunneling Applications Fast Switching Oscillation Amplification

7 SQUIDS Josephson Junction: Insulating barrier between two superconducting regions Cooper pairs of electrons tunnel across Josephson Junction barrier. Tunneling rate dependent on magnetic flux. Magnetic flux quantum.! 0 = h 2e = 2.067"10#15 T m 2

8 Magnetic Flux modulates current through Josephson Junctions SQUIDS

9 SQUIDS DC operation Current made to flow around the loop through both Josephson junctions. Electrons tunnel through the junctions and interfere. Magnetic field through the loop causes a phase difference between electrons, affects current through the loop. Flux (magnetic field) through the loop induces a current around the loop. This affects the current flowing through the loop, because the net current through each junction is no longer the same. Resulting potential difference across the loop can be measured.

10 SQUIDS Sensitivity L

Magnetic field of brain ~ 10-13 T Magnetoencephalography

11 SQUIDS Applications Very sensitive magnetometer Threshold for squid ~ T Magnetic field of heart ~ T Magnetic field of brain ~ T Magnetoencephalography Ultra Low Field Magnetic Resonance Imaging Magnetic Monopole Searches

12 The removal of an electron from a given kind of metal requires a certain minimum amount of energy: The work function. Metal s electron reside in a potential well. Potential step no tunneling. But some electrons may escape thermally If a positive electrode is brought near the metal s surface the potential step can be changed into a barrier tunneling possible. Field Emission

13 Scanning Tunneling Microscope Barrier formed by separation of tip from sample Tunneling Current highly dependent on separation. Current then measures distance from tip to sample Sensitivities enable atoms to be resolved

14 Coulomb Repulsion and Quantum Tunneling Effect on Stellar Fusion Range of strong nuclear force is approximately 1fm=10-15 m Coulomb repulsion barrier until d=1fm V c =3.43 Mev between protons Classically the kinetic energy of proton must exceed this barrier potential Quantum mechanical tunneling effectively lowers coulomb barrier. Proton must approach approximately within one Debroglie wavelength of the target Quantum_tunneling

15 Coulomb Repulsion and Quantum Tunneling Classically proton would need to climb full coulomb repulsion barrier to get close enough (~1fm) for strong nuclear potential to become effective. Quantum Mechanical barrier penetration allows the proton to tunnel close enough Lowers temperature at which fusion occurs

16 Particle-Wave Propagation Plane Wave: A plane wave is not the most realistic matter wave. Well defined wave number (momentum) but not localized in space. Wave Packet: A reasonably compact moving particle is better represented by a traveling wave pulse. A wave packet (pulse) can be formed from a sum of plane waves of varying wave number. Phase Velocity: Each constituent plane wave moves at its own phase speed v=ω/k. Group Velocity: The speed of the region where the probability density is largest the speed of the particle, or group velocity-may differ distinctly from the speeds of constituent plane waves. Dispersion: An initially welllocalized wave pulse will spread out with time as its constituent plane waves get progressively out of step.

17 A simple wave group Wave group of just two plane waves can be written as:!(x,t) = Ae i(k 1x!" 1 t) + Ae i(k 2x!" 2 t) Choose k 1 and k 2 and ω 1 and ω 2 such that: k 1 = k 0 + dk k 2 = k 0! dk! 1 =! 0 + d!! 2 =! 0! d! Thus!(x,t) = Ae i[(k 0+dk)x!(" 0 +d" )t] + Ae i[(k 0!dk)x!(" 0!d" )t]!(x,t) = Ae i(k 0x!" t) 0 (e +i[(dk)x!(d" )t] + e!i[(dk)x!(d" )t] )!(x,t) = Ae i(k 0x!" t) 0 2cos[(dk)x! (d")t] The Probability density becomes! * (x,t)!(x,t) = 4A 2 cos[(dk)x! (d")t] And the phase velocities disappear!!! Group velocity dω/dk dictates motion of wave packet

18 A Particle-like wave Most general expression for a wave group is +"!(x,t) = #!" A(k)e i(kx!"t) dk Integral over all wavenumbers. Each weighted by factor A(k). Dispersion relations: Plane wave solutions to Maxwell s Equations in vacuum: (E=cp) ΕΜ wave dispersion relation ω(k)=ck Vphase =λf=ω(k)/k=c Free particle Schroedinger equation solutions are E=p 2 /2m Matter wave dispersion relation ω(k)=hk 2 /[(2π)2m] Vphase =λf=ω(k)/k=hk/[(2π)2m]

19 A Particle-like wave: Gaussian Wave Packet A Gaussian wave packet at time t=0 may be written as!(x, 0) = Ce!(x/2" )2 e ik 0x A right-moving plane wave multiplied by a Gaussian bump Not infinitely broad in space. Momentum not perfectly defined. The oscillatory exponential does give it an approximate (mean) wave number and momentum hk 0 /2π. Probability density at any given time is!(x,t) 2 = C D 2 t 2 / 4" exp[!(x! st) 4 2" 2 (1+ D 2 t 2 / 4" 4 ) ] s! d!(k) dk k0 D! d 2!(k) dk 2 k0

20 A Particle-like wave: Gaussian Wave Packet The probability density at time t=0 is then!(x, 0) 2 = C 2!(x/2" )2 e If D=0 pulse is just sliding along the x-axis at speed s.!(x!st) 2!(x,t) 2 = C 2 2" e 2 With group velocity v group = s = d!(k) dk k0 Group velocity is the same even for non-zero D. D governs how the probability density spreads out dispersion. ΕΜ waves: v group = s = d!(k) dk Matter waves: v group = s = d!(k) k0 = d dk dk k0 = d dk ck k 0 = c!k 2 2m k 0 =!k 0 m = 2v phase Note that v group =2 v phase!!!!!!!

21 Dispersion- Phase Velocity/Group Velocity

22 Quantum Mechanics and Wave-Particle Duality Fourier Wave Packets Uncertainty Principle

23 Uncertainty Principle Uncertainty Principle

24 Electromagnetic Pulse in a medium with wavelength dependent refractive index Earth s ionosphere refractive index for electromagnetic waves used in GPS n(!) = 1! b v = c phase! 2 1! b! 2 Dispersion relation! k = Group velocity c v group = d! dk k 0 = v phase (! 0 ) = 1! b! 2!(k) = b + (kc) k 0 c 2 b + (k 0 c) 2 = c 1! b /! 0 c 1! b! Group velocity ok v group <c But phase velocity??? V phase >c???? Phase carries no information!!!

25 Dispersion Spreading of wave pulse occurs when second derivative of w.r.t k is non-zero D is dispersion coeffiecient Dispersion occurs for EM waves in non-linear media. Dispersion occurs for matter waves even in vacuum!(x,t) 2 = C D 2 t 2 / 4" exp[!(x! st) 4 2" 2 (1+ D 2 t 2 / 4" 4 ) ]

26 Links %3AQuantum_tunnel_effect_and_its_application_to_the_scanning_tunneling_microscope.ogv

Quantum Mechanical Tunneling

The square barrier: Quantum Mechanical Tunneling Behaviour of a classical ball rolling towards a hill (potential barrier): If the ball has energy E less than the potential energy barrier (U=mgy), then