Quantum Mechanics for Scientists and Engineers. David Miller

Transcription

1 Quantum Mechanics for Scientists and Engineers David Miller

2 Time evolution of superpositions

3 Time evolution of superpositions Superposition for a particle in a box

4 Superposition for a particle in a box Suppose we have an infinitely deep potential well a particle in a box with the particle in a linear superposition for example, with equal parts of the first and second states of the well 1 E1 E2 2 t, expi tsin expi tsin L L L

5 Superposition for a particle in a box Note for each eigenfunction in the superposition it is multiplied by the appropriate complex exponential time-varying function En exp i t This superposition is also normalied 1 E1 E2 2 t, expi tsin expi tsin L L L

6 Superposition for a particle in a box From this superposition 1 E1 E2 2 t, expi tsin expi tsin L L L t, 2 we can multiply it by its complex conjugate to get the probability density sin sin E E 2cos t sin sin L L L L L

7 Superposition for a particle in a box E1 E2 2 exp i tsin exp i tsin L L E1 E2 2 expi tsin expi tsin L L E2E1 E2E1 sin sin sin sin exp i t exp i t L L L L sin sin E E 2cos t sin sin L L L L multiplied by its complex conjugate

8 Superposition for a particle in a box t, 2 Note this probability density sin sin E E 2cos t sin sin L L L L L has a part that is oscillating in time at an angular frequency 21 E2 E1 / 3 E1 / Note also that the absolute energy origin does not matter here for this measurable quantity only the energy difference E E matters 2 1

9 Particle in a box As a reminder here are the first few particlein-a-box energy levels and their associated wavefunctions plotted with the orange dashed lines as horiontal axes Energy n 3 n 2 n 1 E 3 E 2 E 1

10 Superposition The n 1 spatial eigenfunction 1 is plotted here with the bottom of the box as its horiontal axis n 1 Wavefunction 1

11 Superposition For the probability density 2 1 note the different shape Multiplying by the time dependent factor gives E1 1, texp i t 1 The probability densities are the same, t n 1 Probability density 2 1 t 2 1,

12 Superposition Similarly The n 2 spatial eigenfunction 2 is plotted here with the bottom of the box as its horiontal axis n 2 Wavefunction 2

13 Superposition The probability density 2 2 is a positive function Multiplying by the time dependent factor gives E2 2, texp i t 2 The probability densities are the same, t n 2 Probability density 2 2 t 2 2,

14 Superposition An equal superposition of the two oscillates at the angular frequency E E / 3 E / , t, t, t E2 E1 t 1 2 2cos Probability density, t, t 2 1 2

15

16 Time evolution of superpositions Superposition for the harmonic oscillator

17 Superpositions and oscillation Quite generally if we make a linear combination of two energy eigenstates with energies E a and E b the resulting probability distribution will oscillate at the (angular) frequency E E / ab a b

18 Superpositions and oscillation So, if we have a superposition wavefunction, exp E E a b r ab t c a i t a cb exp i t b r r then the probability distribution will be 2 2 r, t c r c r ab a a b b Ea Eb t 2 c rc r cos where arg ab caa r c b b r a a b b ab

19 Harmonic oscillator As a reminder here are the first two harmonic energy levels and their associated wavefunctions plotted with the orange dashed lines as horiontal axes Energy 3 /2 /2

20 Superposition The n 0 spatial eigenfunction 0 is plotted here with the bottom of the parabolic well as its horiontal axis n 0 Wavefunction 0

21 Superposition For the probability density 0 2 note the narrower shape Multiplying 0 by the time dependent factor gives E0 0, texp i t 0 The probability densities are the same, t n 0 Probability density 2 0

22 Superposition The n 1 spatial eigenfunction 1 is plotted here with the bottom of the parabolic well as its horiontal axis n 1 Wavefunction 1

23 Superposition For the probability density 2 1 note it is positive Multiplying by the time dependent factor gives E1 1, texp i t 1 The probability densities are the same, t n 1 Probability density 2 1

24 Superposition An equal superposition of the two oscillates at the angular frequency E E, t, t, t 2cos t / Probability density

25 Superposition An equal superposition of the two oscillates at the angular frequency E E, t, t, t 2cos t / Probability density, t, t 2 0 1

26

27 Time evolution of superpositions The coherent state

28 The coherent state The coherent state for a harmonic oscillator of frequency is 1 N, tcnnexpin t n n0 2 where n N expn cnn n! and the n are the harmonic oscillator eigenstates

29 The coherent state Incidentally, note that for the expansion coefficients c Nn n 2 N expn cnn n! This is the Poisson distribution from statistics with mean N and standard deviation N We will make no direct use of this here but in the end it explains, e.g., the Poissonian distribution of photons in a laser beam

30 Coherent state N, t 2 N 1 Coherent state oscillations with N, t n0 1 cnn exp in t n 2 c Nn N n expn n!

31 Coherent state N, t 2 N 3 Coherent state oscillations with N, t n0 1 cnn exp in t n 2 c Nn N n expn n!

32 Coherent state N, t 2 N 10 Coherent state oscillations with N, t n0 1 cnn exp in t n 2 c Nn N n expn n!

33 Coherent state N, t 2 N 100 Coherent state oscillations with N, t n0 1 cnn exp in t n 2 c Nn N n expn n!

34 Finite well superposition Make an equal superposition of the first three states of a finite potential well as in our previous example Because the energies are not rationally related the superposition never repeats e.g., in the probability density in time Vo 8 E

35