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Chapter 12 Free Particle and Tunneling Chapter 12 Free Particle and Tunneling P. J. Grandinetti Chem. 4300 P. J. Grandinetti Chapter 12: Free Particle and Tunneling Bound or free particle in classical mechanics? ̇ Etotal = K.x/ + V.x/ P. J. Grandinetti Chapter 12: Free Particle and Tunneling Bound or free particle in quantum mechanics? Back to Schrödinger equation: [ ] )Ψ(x; t/ E Ψ(x; t/ = K̂ + V̂ .x/ Ψ(x; t/ = i` «­­­¯­­­¬ )t Ĥ We divide the potentials, V.x/, into two groups 1 those that bind a particle to a particular region of space 2 those that do not bind a particle to a particular region of space We saw example of 1st with particle in infinite well in last chapter. As example of 2nd, imagine particle moving from left to right with constant momentum and no forces acting on it—a free particle. P. J. Grandinetti Chapter 12: Free Particle and Tunneling A Quantum Free Particle Let’s define a free particle as having E > 0 and V̂ .x/ = 0, 0 2 ̂ ̂ > ̂ `2 d .x/ H .x/ = .V.x/ + K/ .x/ = * = E .x/ 2m dx2 .x/ is eigenstate of Ĥ with stationary state wave function Ψ(x; t/ = .x/e*iEt_` Wave function for free particle traveling left to right with p = `k and E = `! Ψ(x; t/ = Aei.kx*!t/ = Aeikxe*iEt_` where ù p 2mE and ikx right traveling particle k = ` = ` .x/ = Ae For particle traveling right to left we use p = *`k and have .x/ = Ae*ikx left traveling particle P. J. Grandinetti Chapter 12: Free Particle and Tunneling Interesting thing about this free particle wave function: .x/ = Aeikx There is no uncertainty in its momentum. ∞ ∞ ê ë < ̂ ` < d .x/ Calculate expectation value: p = Ê .x/ p .x/ dx = *i Ê .x/ dx −∞ −∞ dx Substituting right traveling particle, .x/ = Aeikx, gives ( ) ∞ d eikx ∞ ∞ ê ë ` | |2 *ikx ` | |2 *ikx ikx ` < ` p = *i Ê A e dx = k Ê A e e dx = k Ê .x/ .x/dx = k −∞ dx −∞ −∞ Calculate expectation value for p̂2 from .x/, ∞ d2.eikx/ ∞ ∞ êp2ë = .*i`/2 |A|2e*ikx dx = .*i`/2.ik/2 |A|2e*ikxeikxdx = `2k2 <.x/ .x/dx = `2k2 Ê 2 Ê Ê −∞ dx −∞ −∞ ù ù Uncertainty in momentum is Δp = êp2ë *.êpë/2 = `2k2 *.`k/2 = 0 Since ΔxΔp g `_2 then Δx = ∞, momentum precisely known, but no idea where particle is. P. J. Grandinetti Chapter 12: Free Particle and Tunneling Another thing about this free particle wave function Did you noticed a small problem with this free particle wave function? It can’t be normalized. ∞ < : Ê .x/ .x/dx = ∞ −∞ We have an “end effect”, or better stated a “no-end effect.” Could solve this problem with traveling wave packet, but then rest of math would get more difficult while we gain little new physical insight. Keeping this caveat in mind we’ll continue working with an un-normalizable free particle wave function and assume it does not seriously affect our conclusions. P. J. Grandinetti Chapter 12: Free Particle and Tunneling Free particle approaches a step potential P. J. Grandinetti Chapter 12: Free Particle and Tunneling Free particle approaches a step potential Imagine conduction electron in metal moving towards surface of metal. e– metal Inside metal electron feels attraction to positively charged metal nuclei. Classical particle must overcome attractive potential, V<0, to escape from surface 0 if x < 0 Take approximate potential energy function of V.x/ = > V0 if x 0 P. J. Grandinetti Chapter 12: Free Particle and Tunneling Two cases when free particle approaches a step potential case b case a We consider 2 separate cases: < (a) E V0 g (b) E V0 Let’s examine wave function for quantum particle with total energy E in each case. P. J. Grandinetti Chapter 12: Free Particle and Tunneling Free particle approaches a step potential case a < Case (a) E V0 P. J. Grandinetti Chapter 12: Free Particle and Tunneling < Free particle approaches a step potential, Case: (a) E V0 Start by breaking total wave function into 2 parts: Wave function in metal, in Wave function outside metal, out ̂ casê a Left of x = 0 where V.x/ = 0 Right of x = 0, where V.x/ = V0 2 d .x/ 2 in 2mE d out.x/ 2m.V0 * E/ + in.x/ = 0 * .x/ = 0 dx2 `2 2 `2 out «¯¬ dx «­­­­¯­­­­¬ 2 > ; k 0 real k2 > 0; real 1 2 Has general solution Has general solution ik1x *ik1x f .x/ = Ae + Be for x 0 k2x *k2x g in out.x/ = Ce + De for x 0 ù ù where k = 2mE_` ` 1 where k2 = 2m.V0 * E/_ P. J. Grandinetti Chapter 12: Free Particle and Tunneling < Free particle approaches a step potential, Case: (a) E V0 At x = ∞ the C term goes unphysically to ∞. Setting C = 0 leaves *k2x out.x/ = De Where in and out meet at x = 0 wave function must be finite, single valued, and continuous. d .0/ d .0/ .0/ = .0/ and in = out in out dx dx Gives 2 equations, A + B = D and ik1A * ik1B = *k2D Taking sum and difference of 2 equations gives 0 1 0 1 D ik D ik A = 1 + 2 and B = 1 * 2 2 k1 2 k1 P. J. Grandinetti Chapter 12: Free Particle and Tunneling < Free particle approaches a step potential, Case: (a) E V0 < Wave function when E V0 is given by 0 1 0 1 h D ik D ik 2 ik1x 2 *ik1x f n 1 + «¯¬e + 1 * «¯¬e x 0 (in) 2 k1 2 k1 n «­­­­­­¯­­­­­­¬ ⟶ «­­­­­­¯­­­­­­¬ ⟵ .x/ = l n A B n j De*k2x x g 0 (out) particle penetration into classically excluded region probability P. J. Grandinetti Chapter 12: Free Particle and Tunneling < Free particle approaches a step potential, Case: (a) E V0 The total wave function can be written h ` ` i.k1x*Et_ / i.*k1x*Et_ / f n Ae + Be x 0 (in) Ψ(x; t/ = l ` n *k2x *iEt_ g j De e x 0 (out) f We recognize two terms in in (when x 0) as corresponding to right and left traveling waves. P. J. Grandinetti Chapter 12: Free Particle and Tunneling < Free particle approaches a step potential, Case: (a) E V0 Aeik1x is incident wave coming from inside metal towards surface Be*ik1x is reflected wave traveling back into metal. We can define reflection coefficient, R, as 0 1 0 1 0 1 0 1 ik < ik ik ik 1 * 2 1 * 2 1 + 2 1 * 2 B<B k k k k R = = 0 1 1 0 1 1 = 0 1 1 0 1 1 = 1 A<A ik < ik ik ik 1 + 2 1 + 2 1 * 2 1 + 2 k1 k1 k1 k1 R = 1 means total reflection and electron doesn’t escape metal just as we expected for classical free particle. P. J. Grandinetti Chapter 12: Free Particle and Tunneling < Free particle approaches a step potential, Case: (a) E V0 But, electron is wave, so how far away from metal surface does wave go? Look at probability ( ù ) *2x_ `_ 2m.V *E/ < < *2k2x < *2x∕Δx | |2 0 out out = D De = D De = D e Normalization issues aside, if we take Δx = 1_k2 as the barrier penetration distance, then ` Δx = ù 2m.V0 * E/ Example Measurement of copper work function shows that V0 * E = 4 eV. Estimate distance Δx that electron can penetrate into classically excluded region outside metal block. ` ` Δx = ù = ù ù 1 Å 2m.V0 * E/ 2me.4 eV/ Extends out distance that is roughly diameter of an atom. P. J. Grandinetti Chapter 12: Free Particle and Tunneling Free particle approaches a step potential case b g Case (b) E V0 P. J. Grandinetti Chapter 12: Free Particle and Tunneling g Free particle approaches a step potential, Case: (b) E V0 Consider situation when total energy of particle exceeds V0. Recall F = *dV_dx. Effect of changing potential is to exert a force on particle. Step potential will exert an impulsive force on a particle. In this case impulsive force will slow down particle but won’t stop it from continuing to travel into positive x region. P. J. Grandinetti Chapter 12: Free Particle and Tunneling g Free particle approaches a step potential, Case: (b) E V0 Particle kinetic energy is ... p2 p2 E = in for x < 0 .E * V / = out for x > 0 2m 0 2m Schrödinger equation for 2 regions would be 2 d in.x/ 2mE d2 .x/ 2m.E * V / + .x/ = 0 out 0 2 `2 in + out.x/ = 0 dx dx2 `2 oscillatory solution for .x/: out oscillatory solution for in.x/: ik2x *ik2x g out.x/ = Ce + De for x 0 ik1x *ik1x f in.x/ = Ae + Be for x 0 where where ù ù k = 2mE_` = p _` ` ` 1 in k2 = 2m.E * V0/_ = pout_ P. J. Grandinetti Chapter 12: Free Particle and Tunneling g Free particle approaches a step potential, Case: (b) E V0 case b ik1x *ik1x f ik2x *ik2x g in.x/ = Ae + Be for x 0 out.x/ = Ce + De for 0 Particle traveling from x < 0 is allowed to pass through x = 0 and head into x > 0. Nothing in our model causes it to return. To account for this we set D = 0. Can’t assume particle traveling in x < 0 towards x = 0 will make it past x = 0. < > Particle might be reflected, like E V0 case, even though this is E V0 case, where classical free particle would never be reflected. P. J. Grandinetti Chapter 12: Free Particle and Tunneling g Free particle approaches a step potential, Case: (b) E V0 Require that in and out meet at x = 0 with d .0/ d .0/ .0/ = .0/ and in = out in out dx dx This gives 2 equations, A + B = C and k1.A * B/ = k2C Solving for B and C in terms of A gives k * k 2k B = 1 2 A and C = 1 A k1 + k2 k1 + k2 P.
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