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. Content  Superposition of states and particle motion  Time dependence of wavefunctions and states  Barrier Penetration and Tunneling
3. 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
4. Time-dependent SEQ  To explore how particle wavefunctions evolve with time, which is useful for a number of applications as we shall see, we need to consider the time-dependent SEQ: 22dYY(,)(,) x t d x t This equation describes the full U()(,) x Y x t i time- and space dependence of 2m dx2 dt a quantum particle in a potential U(x), replacing the classical i2 = -1 particle dynamics law, F=ma  Important feature: Superposition Principle  The time-dependent SEQ is linear in Y (a constant times Y is also a solution), and so the Superposition Principle applies: If Y1 and Y2 are solutions to the time-dependent SEQ, then so is any linear combination of Y1 and Y2 (example: Y 0.6 Y1 + 0.8iY2)
5. 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.
6. Complex Wavefunctions  How can imaginary numbers describe a physical system? Y ( x,t ) Aei( kx t ) Wavefunction of a free particle with momentum p = ħk and energy E = ħ  What we would measure is in the ‘square’ of Y(x,t): namely, the probability distribution. Is it real for this wavefunction? 2 2  For a complex wavefunction, Probability equals (absolute value) = |Y| = Y*Y , where Y* is the complex conjugate of Y. (replace i with –i) Y *Y Ae i( kx t ) Aei( kx t ) A2 A real constant.  We find that an unconfined free particle with momentum ħk has an equal probability of being anywhere on the x-axis. Of course, if we have the particle in our macroscopic apparatus of dimension L, then the constant A is roughly 1/L1/2 in order that Y * Y dx = 1.
7. 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
8. Time-dependence of Eigenstates y  Example 2: Time-evolution of an “eigenstate” x 0 L  An “eigenstate” y is described by a single E, so we can write: 22dYY(,)(,) x t d x t U()(,)(,) x Y x t E Y x t i 2m dx2 dt E This equation has the solution: Y(x, t) y(x)e it with   This wavefunction has a “complex” time-dependence. But, we are mostly interested in what we measure, |Y(x,t)|2: Y(x,t)22 y* (x)e i  t y (x)e i  t y (x) As previously stated, the probability density |Y(x,t)|2 associated with eigenstates of the SEQ doesn’t change with time. Thus the name for states with well-defined energies Stationary States
9. 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
10. Homework # 5  How many experiments can you recall that support the wave theory of light? The particle theory of light? The wave theory of matter? The particle of theory of matter?  Discuss similarities and differences between a matter wave and an electromagnetic wave?