Lecture physics a2 superposition time dependent quantum states huynh quang linh

Lecture 8 Superposition & Time Dependent Quantum States x |y(x,t)|2 U= U= 0 xL Content  Superposition of states and particle motion  Time dependence of wavefunctions and states  Barrier Penetrati[.]

Lecture 8: Superposition & Time-Dependent Quantum States U= |y(x,t)|2 U= x L x Content  Superposition of states and particle motion  Time dependence of wavefunctions and states  Barrier Penetration and Tunneling 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”  d y ( x)   U ( x)y ( x)  Ey ( x) 2m dx “Functions that fit”: (l = 2L/n) “Doesn’t fit”: y(x) y(x) U= n=1 n=3 U= L n=2 x L x 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: d Y ( x, t ) d Y ( x, t )  U ( x ) Y ( x , t )  i 2m dx dt 2 i2 = -1  This equation describes the full time- and space dependence of a quantum particle in a potential U(x), replacing the classical 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) Motion of a Free Particle  Example #1: Wavefunction of a free particle  A free particle moves without applied forces; so we set U(x) = d Y ( x, t ) d Y ( x, t )   i 2m dx dt Traveling wave solution Y ( x ,t )  Aei( kxt ) Prove it Take the derivatives: dY  ik Aei ( kx t ) dx d 2Y  (ik ) Aei ( kx t )   k Aei ( kx t ) dx dY  (i )Aei ( kx t ) dt 2k   2m i  1 Wavefunction of free particle p2 classicall y, E 2m with p  momentum and E  kinetic energy From DeBroglie, p = h/l = ħk Now we see that E = ħ = hf These relations provide the correspondence between particle and wave pictures 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? For a complex wavefunction, Probability equals (absolute value)2 = |Y|2 = 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 = 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 that the required spread in k-vectors (and by p = ħk, momentum states, is determined by the Heisenberg Uncertainty Principle: Dp·Dx ≈ ħ ...Content  Superposition of states and particle motion  Time dependence of wavefunctions and states  Barrier Penetration and Tunneling Time- independent SEQ  Up to now, we have considered quantum. .. n=2 x L x 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:... the full time- and space dependence of a quantum particle in a potential U(x), replacing the classical particle dynamics law, F=ma Important feature: Superposition Principle  The time- dependent