Introduction to Quantum Mechanics

Size: px

Start display at page:

Download "Introduction to Quantum Mechanics"

Derrick Allen
5 years ago
Views:

1 Introduction to Quantum Mechanics INEL Solid State Devices - Spring 2012 Manuel Toledo January 23, 2012 Manuel Toledo Intro to QM 1/ 26

2 Outline 1 Review Time dependent Schrödinger equation Momentum operator Time-independent form of SE Free Particle 2 Electron Meeting a Potential Barrier 3 Tunneling 4 The Quantum Well Manuel Toledo Intro to QM 2/ 26

3 Time dependent Schrödinger equation Schrödinger equation Time dependent Schrödinger equation (SE): i Ψ t = 2m 2 Ψ+U(x,y,z)Ψ Manuel Toledo Intro to QM 3/ 26

4 Momentum operator Momentum operator r can be x, y or z. i r Expectation value of the momentum in the x direction: Energy operator: < p x >= Ψ V i i t Ψ x dv Manuel Toledo Intro to QM 4/ 26

5 Time-independent form of SE Time-independent form of SE 2 ψ + 2m (E U(x,y,z))ψ = 0 2 Manuel Toledo Intro to QM 5/ 26

6 Free Particle Free Particle Consider a particle of mass m in a 1D space that is isolated - infinitely far from other particles has total particle energy is E experiences no forces ( F = U = 0 ) and thus has constant potential energy (chosen to be 0). From the POV of QM to characterize the particle one must solve the time-independent Schrödinger equation d 2 ψ dx 2 + 2mE 2 ψ = 0 (1) Manuel Toledo Intro to QM 6/ 26

7 Free Particle For free particle, Ψ(x,t) = Ae i(kx E t) +Be i(kx+e t) The free particle wave-function is similar to a traveling wave. Manuel Toledo Intro to QM 7/ 26

8 Free Particle For free particle, Ψ(x,t) = Ae i(kx E t) +Be i(kx+e t) The free particle wave-function is similar to a traveling wave. If the particle moves in the x direction, B = 0. If it moves in the x direction, A = 0. Since, for a particle moving in the +x direction, ψ ψ = A A = constant for all values of x, one has equal probability of finding the particle anywhere. Manuel Toledo Intro to QM 7/ 26

9 Free Particle Momentum The particle s momentum is < p x > = = k ψ i dψ dx dx ψ ψdx = k = h λ = 2mE in agreement with classical mechanics. Manuel Toledo Intro to QM 8/ 26

10 Free Particle Wave-packet Example illustrating wafe-function superposition. (Taken from Craig Casey, 1999). Manuel Toledo Intro to QM 9/ 26

11 Potential barrier Consider a 1D potential with the shape shown below. U e - U=V 2 U=0 An electron approaching a potential barrier. We must now solve two equations. For x 0 the electron is free (U = 0), and the solution is the one found for a free particle: ψ = Ae ikx +Be ikx Manuel Toledo Intro to QM 10/ 26

12 For x > 0, U = V 2 and the SE becomes 2 ψ x 2 + 2m 2 (E V 2)ψ = 0 ψ = Ce ik 2x +De ik 2x where k 2 = 2m(E V 2) Manuel Toledo Intro to QM 11/ 26

13 and At x = 0, ψ must be continuous ψ(0 ) = ψ(0+) A+B = C +D Now consider a particle incident from the left. For this case, we can set D = 0 and get that A+B = C Because ψ x is also required to be continuous, ika ikb = ik 2 C or k(a B) = k 2 C, which can be expressed as C = k k 2 (A B). Manuel Toledo Intro to QM 12/ 26

14 Substituting in the previous expression yields A+B = k (A B) k ) 2 ) A (1 kk2 = B (1+ kk2 A k k 2 = B k +k 2 k 2 k 2 B A = k k 2 k +k 2 and C A = 2k k +k 2 Manuel Toledo Intro to QM 13/ 26

15 Observations: If E > V 2, both k and k 2 are real; solutions are oscillatory; B A = k k2 k+k 2 is finite! Thus, there is a finite probability that the barrier will reflect back the electron, causing it to turn back to the left. If E < V 2, k 2 = 2m(E V2) is imaginary; the wavefunction declines exponentially for x > 0. C A = 2k k+k 2 > 0 so there is a non-zero probability for the electron to penetrate the barrier! Manuel Toledo Intro to QM 14/ 26

16 Tunneling Consider an electron approaching the potential barrier of finite width: U e - U=V 2 U=0 W An electron approaching a finite-width potential barrier. Manuel Toledo Intro to QM 15/ 26

17 for x < 0 and for x > W ψ = free electron. for 0 x W, ψ = ψ 2 for a potential barrier. Thus, A + e ikx +A e ikx x < 0 ψ(x) = B + e ik2x +B e ik 2x 0 x W C + e ikx +C e ikx x > W where k = 2mE and k 2 = 2m(E V 2 ). Manuel Toledo Intro to QM 16/ 26

18 If an electron approaches the barrier from the left, C = 0. Rather than proceeding to match the wave-functions and its derivative at x = 0 and x = W to determine the relation between coefficients explicitly, we will just express the result in terms of the transmission coefficient, defined as T = C + 2 A + Manuel Toledo Intro to QM 17/ 26

19 For E > V 2, T = 1+ 1 V 2 2 4E(E V 2 ) S2 where S = sin(k 2 W). Notice that this equals unity for any barrier width for which k 2 W = nπ, where n is any integer. For E < V 2, T = where S = sinh(ik 2 W) V 2 2 4E(V 2 E) S2 h Manuel Toledo Intro to QM 18/ 26

20 ψ for the tunneling example. Observe that an electron with E < V 2 have a finite probability of penetrating the barrier. W Manuel Toledo Intro to QM 19/ 26

21 Quantum Well U V W x A one-dimensional quantum well. Manuel Toledo Intro to QM 20/ 26

22 Trial wave-functions: A 1 e αx +B 1 e αx x < 0 ψ(x) = A 0 sin(kx)+b 0 cos(kx) 0 x W A 2 e αx +B 2 e αx x > W where and 2m(V1 E) α = 2mE k = Manuel Toledo Intro to QM 21/ 26

23 Observe that if E > V 1, α is imaginary and the corresponding exponential terms in the wave-function are oscillatory. if E < V 1, ψ should decay as it penetrates the barrier, and vanish far from it. This behavior requires that B 1 = A 2 = 0 The resulting wave-function is, A 1 e αx x < 0 ψ(x) = A 0 sin(kx)+b 0 cos(kx) 0 x W B 2 e αx x > W Manuel Toledo Intro to QM 22/ 26

24 Continuity boundary conditions at x = 0 and x = W: ψ(0 ) = ψ(0+) ψ(w ) = ψ(w+) Solution is: dψ dψ (0 ) = dx dx (0+) dψ dψ (W ) = dx dx (W+) A 1 = B 0 B 0 = k α A 0 A 0 sin(kw)+b 0 cos(kw) = B 2 e αw k(a 0 cos(kw) B 0 sin(kw)) = αb 2 e αw Manuel Toledo Intro to QM 23/ 26

25 Solving B 2, substituting and rearranging: A 0 (kcos(kw) +αsin(kw)) B 0 (ksin(kw) αcos(kw)) = 0 After eliminating A 0 and B 0, tan(kw) = 2kα k 2 α 2 In terms of a normalized energy ξ = E/V 1, and using kw = 2mE = W 2mV 1 E V 1 = Wa ξ where a = 2mV1, and that 2kα k 2 α 2 = 2 ξ(1 ξ) 2ξ 1 Manuel Toledo Intro to QM 24/ 26

26 The solution becomes Observe that: tan(wa ξ) = 2 ξ(1 ξ) 2ξ 1 The tangent of θ increases from 0 at θ = 0 to + at θ = π/2; it then jumps discontinuously to and increases again to 0 at θ = π. This behavior repeats with a period π. Manuel Toledo Intro to QM 25/ 26

27 Comparison of the two sides of the result obtained for the quantum well, for Wa = 4π. The two expressions intercept at four energy levels, for which the particle is confined in the well. Manuel Toledo Intro to QM 26/ 26

Final Exam. Tuesday, May 8, Starting at 8:30 a.m., Hoyt Hall.

Final Exam. Tuesday, May 8, Starting at 8:30 a.m., Hoyt Hall. Final Exam Tuesday, May 8, 2012 Starting at 8:30 a.m., Hoyt Hall. Summary of Chapter 38 In Quantum Mechanics particles are represented by wave functions Ψ. The absolute square of the wave function Ψ 2