Physics A2 - Lecture 8: Superposition & Time-Dependent Quantum States - Huynh Quang Linh

Nội dung text: Physics A2 - Lecture 8: Superposition & Time-Dependent Quantum States - Huynh Quang Linh

1. Lecture 8: Superposition & Time-Dependent Quantum States |y(x,t)|2 U= U= x 0 L x
2. Time-independent SEQ Up to now, we have considered quantum particles in “stationary states,” and have ignored their time dependence Remember that these special states were associated with a single energy (from solution to the SEQ) “eigenstates” 2 d 2 y (x) − +U (x)y (x) = Ey (x) 2m dx2 “Functions that fit”: (l = 2L/n) “Doesn’t fit”: y(x) y(x) U= U= n=1 n=3 0 L x 0 L x n=2
3. Motion of a Free Particle Example #1: Wavefunction of a free particle. A free particle moves without applied forces; so we set U(x) = 0. 22dYY(,)(,) x t d x t −=i i = −1 2m dx2 dt Traveling wave solution Y ( x,t ) = Aei( kx−t ) Wavefunction of free particle p2 Prove it. Take the derivatives: classicall y, = E 2k 2 2m dY =  with p = momentum = ik Aei(kx −t) 2m dx and E = kinetic energy 2 d Y 2 i(kx −t) 2 i(kx −t) = (ik) Ae = −k Ae From DeBroglie, p = h/l = ħk. dx2 Now we see that E = ħ = hf dY = (−i)Aei(kx −t) dt These relations provide the correspondence between particle and wave pictures.
4. FYI: Wavepackets The plane-wave wavefunction for a particles is a rather extreme view: Y ( x,t ) = Aei( kx−t ) It describes a particle with well defined momentum, p = ħk, but completely uncertain position. By adding together (“superposing”) waves with a range of wave vectors Dk, we can produce a localized wave packet. We can imagine such a packet in space: Dx We saw in Lecture 6 that the required spread in k-vectors (and by p = ħk, momentum states, is determined by the Heisenberg Uncertainty Principle: Dp·Dx ≈ ħ
5. Time-dependence of Superpositions It is possible that a particle can be in a superposition of “eigenstates” with different energy. Because superpositions are also solutions of the time-dependent SEQ! What does it mean that a particle is “in two states”. What is its E? To answer this, let’s see how superpositions evolve with time? Consider a simple example using our trusty “particle in an infinite square well” system: A particle is described by a wavefunction involving a superposition of the two lowest infinite square well states (n=1 and 2) Y(x) −−i t i t 12U= U= Y(,)x t = yy12 () x e + () x e E E y1  = 1  = 2 1  2  h2 E = E = 4 E 0 L x 1 8mL2 2 1 y2