“All of modern physics is governed by that magnificent and thoroughly confusing discipline called quantum mechanics It has survived all tests and there is no reason to believe that there is any flaw[.]
“All of modern physics is governed by that magnificent and thoroughly confusing discipline called quantum mechanics It has survived all tests and there is no reason to believe that there is any flaw in it….We all know how to use it and how to apply it to problems; and so we have learned to live with the fact that nobody can understand it.” Murray Gell-Mann Lecture 9: Barrier Penetration and Tunneling nucleus x U(x) U(x) U0 L E A B x C B x A Content How quantum particles tunnel Nuclear Decay Solar Fusion NH3 Maser “Leaky” Particles: Revisited Due to “barrier penetration”, the electron density of a metal actually extends outside the surface of the metal! x Vo Work function F EF Occupied levels x = Assume that the work function (i.e., the energy difference between the most energetic conduction electrons and the potential barrier at the surface) of a certain metal is F = eV Estimate the distance x outside the surface of the metal at which the electron probability density drops to 1/1000 of that just inside the metal ( x) (0) e Kx 1000 using K 2m2 e V0 E 2 2m2 e F 2 h x ln 0.3nm K 1000 5eV 1 11.5 nm 1.505eV nm2 Application: Tunneling Microscopy Due to the quantum effect of “barrier penetration,” the electron density of a material extends beyond its surface: One can exploit this to measure the electron density on a material’s surface: material x Metal tip STM tip ~ nm material STM tip Real STM tip DNA Double Helix: Na atoms on metal: STM images www-aix.gsi.de/~bio http://www.quantum-physics.polytechnique.fr/en/ Tunneling Through a Barrier (1) U(x) What is the “Transmission Coefficient T”, the probability an incident particle tunnels through the barrier? U=Uo Consider a barrier (II) in the middle of a very wide infinite square well U=0 I II III x L To get an “exact” result describing how quantum particles penetrate this barrier, we write the proper wavefunction in each of the three regions shown in Figure: Region I: ( x) A sin kx A cos kx E > U: oscillatory solution I Region II: II ( x) B1e B2e Kx Region III: III ( x) C1 sin Kx E < U: decaying solution kx C2 cos kx E > U: oscillatory solution Next we would need to apply the “continuity conditions” for both and d/dx at the boundaries x = and x = L to determine the A, B, and C coefficients Tunneling Through a Barrier (2) In general the tunneling coefficient T can be quite complicated (due to the contribution of amplitudes “reflected” off the far side of the barrier) U(x) U0 E L However, in many situations, the barrier width L is much larger than the ‘decay length’ 1/K of the penetrating wave; in this case (KL >> 1) the tunneling coefficient simplifies to: T Ge 2KL where E E G 16 1 U0 U0 x 2m U E K G This is nearly the same result as in the “leaky particle” example! Except for G: • G slightly modifies the transmission probability • G arises from the fact that the amplitude at 0 0.25 0.5 0.75 x = is not a maximum E/U0 Tunneling Through a Barrier (3) T Ge where U(x) 2KL U0 E E G 16 1 U0 U0 2m U E K E L x By far the dominant effect is the decaying exponential*: 2 KL T e T depends on the energy below the barrier (U0-E) and on the barrier width L 0.6 The plot illustrates how the transmission coefficient T changes as a function of barrier width L, for two different values of the particle energy E=2/3 U0 T 0.4 E=1/3 U0 0.2 0.5 0.75 1.25 1.5 L *In fact, some references (wrongly) completely omit G (including Phys 214 before 2006!) We will state when you can ignore G Example: Barrier Tunneling in an STM Let’s consider a simple problem: An electron with a total energy of E=6 eV approaches a potential barrier with a height of Uo = 12 eV If the width of the barrier is L=0.18 nm, what is the probability that the electron will tunnel through the barrier? U(x) U0 metal E STM tip L air gap x Example: Barrier Tunneling in an STM Let’s consider a simple problem: An electron with a total energy of E=6 eV approaches a potential barrier with a height of Uo = 12 eV If the width of the barrier is L=0.18 nm, what is the probability that the electron will tunnel through the barrier? T Ge K 2me T 4e U0 E STM tip metal L air gap x E E 1 1 G 16 1 16 1 U0 U0 2 2 2KL U E 2 U(x) 2me 6eV 1 U E 12.6 nm h2 1.505 eV-nm2 2(12.6)(0.18) 4(0.011) 4.3% Question: What will T be if we double the width of the gap? ... “continuity conditions” for both and d/dx at the boundaries x = and x = L to determine the A, B, and C coefficients Tunneling Through a Barrier (2) In general the tunneling coefficient T can.. .Lecture 9: Barrier Penetration and Tunneling nucleus x U(x) U(x) U0 L E A B x C B x A Content How quantum particles... http://www.quantum -physics. polytechnique.fr/en/ Tunneling Through a Barrier (1) U(x) What is the “Transmission Coefficient T”, the probability an incident particle tunnels through the barrier? U=Uo