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FREE PARTICLE GAUSSIAN WAVEPACKET Reading: QM Course packet Wavepacket 1 GAUSSIAN WAVE PACKET - REVIEW • • • • • • Time dependent Schrödinger equation Energy eigenvalue equation (time independent SE) Eigenstates Time dependence (Connection to separation of variables) Mathematical representations of the above Wavepacket 2 Build a "wavepacket" from free particle eigenstates •Ask 2 important questions: •Given a particular superposition, what can we learn about the particle's location and momentum? HEISENBERG UNCERTAINTY PRINCIPLE •How does the wavepacket evolve in time? GROUP VELOCITY Wavepacket 3 We have already discussed the principle of superposition & the time evolution of that superposition in the context of the discrete quantum mechanical states of the infinite potential energy well. We have also already discussed how build a Gaussian wave packet from harmonic waveforms (with a continuous frequency distribution) in the context of the classical rope problem. We now look at the case of superposition of quantum mechanical states of the free particle, which are no longer discrete. Wavepacket 4 Free particle eigenstates y k (x,t) = eikx e -i Ek •Oscillating function •Definite momentum p = hk/2π •Subscript k on w reminds us that w depends on k • What are E and w in terms of given quantities? t k Ek = 2m 2 wk º Ek 2 k2 = 2m 1 0.5 0 j k (x) º yk (x,0) -0.5 y -1 -2 j k (x) = e x 0 2 ikx Wavepacket 5 1 Superposition of eigenstates (Fourier series) 0.5 0 -0.5 j ( x ) = å c kj k ( x ) k k Ek = 2m 2 -2 + 2 -2 + 2 + 2 0 1 0.5 j k (x) = e 2 -1 v phase,k ikx 0 -0.5 wk Ek k = = = k k 2m -1 0 1 0.5 0 -0.5 y 1 -1 0.5 -2 1 x 0 0 0.5 -0.5 0 y -0.5 -1 -2 x 0 2 Wavepacket 6 -1 -2 0 2 1 Superposition of eigenstates (Fourier integral) ¥ 0.5 0 -0.5 j (x) = ò dk A(k)e ikx -1 + -2 -¥ 0 1 2 0.5 0 -0.5 -1 Definite momentum Extended+ position -2 1 0 2 0.5 0 -0.5 y 1 -1 0.5 -2 1 x + 0 2 0 0.5 -0.5 y -1 Localized particle x Indefinite momentum -2 0 0 -0.5 2 Wavepacket 7 -1 -2 0 2 1 Superposition of eigenstates How does it develop in time? 0.5 0 -0.5 ¥ j (x,0) = ò dk A(k)e ikx -1 + -2 -¥ 0 1 2 0.5 ¥ j (x,t) = ò dk A(k)e e ikx -iEk t / 0 -¥ -0.5 -1 Definite momentum Extended+ position -2 k Ek = 2m 2 2 1 0 2 0.5 0 -0.5 y 1 -1 0.5 -2 1 x + 0 2 0 0.5 -0.5 y -1 Localized particle x Indefinite momentum -2 0 0 -0.5 2 Wavepacket 8 -1 -2 0 2 Gaussian wave packet 1 4 2 æ 2a ö -a 2 x2 j (x, 0) = ç e ÷ è p ø 2 2a Localized particle x What is A(k)? We'll ignore overall constants that are not of primary importance (there are conventions about factors of 2π that are important to take care of to get numerical results, but we're after the physics!) A(k) = j k j Projection of general function on eigenstate t =0 Wavepacket 9 Gaussian wave packet 1 4 2 æ 2a 2 ö -a 2 x2 j (x, 0) = ç e ÷ è p ø 2a Localized particle x A(k) = j k j t = 0 = ¥ * dx j ò k (x)j (x,0) -¥ ¥ A(k) = j k j t = 0 @ ò dx e -ikx - a 2 x 2 e -¥ We did this integral (by hand). ¥ using: ò dy e -¥ Wavepacket vy -uy 2 e = p e u v2 4u 10 Gaussian wave packet 1 4 æ 2a 2 ö -a 2 x2 j (x, 0) = ç e ÷ è p ø Localized particle x A(k) = j k j t = 0 @ ¥ ò dx e -ikx - a 2 x 2 e -¥ A(k) @ e Wavepacket - k2 4a 2 11 Gaussian wave packet j (x) @ e 2 -a2 x2 2a Localized particle x A(k) @ e- k 2 4a 2 2 2a k If j(x) is wide, A(k) is narrow and vice versa. Wavepacket 12 j (x) @ e A(k) @ e -a2 x2 - k2 4a 2 A(k) is the projection of j(x) on the momentum eigenstates eikx, and thus represents the amplitude of each momentum eigenstate in the superposition. We need the contribution of a wide spread of momentum states to localize a particle. If we have the contribution of just a few, the Wavepacket 13 location of the particle is uncertain j (x) @ e -2a 2 x 2 2 x -2k 2 4 a 2 A(k) @ e 2 k To define "uncertainty" in position or momentum, we must consider probability, not wave function. Wavepacket 14 j (x) @ e -2a 2 x 2 2 x -2k 2 4 a 2 A(k) @ e 2 Wavepacket 15 k j (x) @ e 2 -2a 2 x 2 x HEISENBERG UNCERTAINTY PRINCIPLE A(k) @ e 2 - 2 k 2 4a 2 Wavepacket 16 k Next, we’ll ask how a general wave packet propagates, And deal with the particular example of the Gaussian wavepacket. In short, we simply attach the exp(-iE(k)t/hbar) factor to each eigenstate and let time run. Difference to non-dispersive equation: not all waves propagate with same velocity. “Packet” does not stay intact! Need to invoke “group velocity” to follow the progress of the “bump”. Wavepacket 17 1 Superposition of eigenstates How does it develop in time? 0.5 0 ¥ j (x,t) = ò dk A(k)e e ikx -iw k t -0.5 Localized particle y -1 -¥ -2 0 x 2 If w depends on k, the different eigenstates (waves) making up this "packet" travel at different speeds, so the feature at x=0 that exists at t=0 may not stay intact at all time. It may stay identifiably intact for some reasonable time, and if it does, how fast does it travel? The answer is "it travels at the group velocity dw/dk" wk º vgroup dw k = = dk m Ek k = 2m Wavepacket 2 w k = k 2m 18 1 Superposition of eigenstates How does it develop in time? 0.5 0 ¥ j (x,t) = ò dk A(k)e e ikx -iw k t -¥ -0.5 Localized particle y -1 -2 0 x 2 The quantity dw/dk may (does!) vary depending on the k value at which you choose to evaluate it. So it must be evaluated at a particular value k0 that represents the center of the packet. The next few pages spend time deriving the basic result. The derivation is not so important. The result is important: vgroup dw = dk k0 k0 = m w k = k 2m Wavepacket 19 For those of you who want more: 1 0.5 ¥ j (x,t) = ò dk A(k)e e ikx -iw k t -¥ 0 -0.5 Localized particle y -1 -2 dw w (k) = w 0 + dk ( k - k0 ) k = k0 j (x,t) = ¥ x 0 2 w is a smooth function of k and w0= w(k0) æ dw -i ç w 0 + dk è ikx ò dk A(k)e e ö ( k - k0 )÷ t ø k=k0 -¥ k - k0 = s Wavepacket 20 For those of you who want more: 1 0.5 0 -0.5 Localized particle y -1 -2 j (x,t) = ¥ ò dk A(k)e æ dw -i ç w 0 + dk è ikx e x ö ( k - k0 )÷ t ø k0 j (x,t) = ò d(k 0 æ dw -i ç w 0 + dk i ( k0 + s ) x è + s ) A(k0 + s)e 2 k - k0 = s -¥ ¥ 0 e ö ( k0 + s - k0 )÷ t ø k0 -¥ j (x,t) = e æ dw -i ç w 0 - k0 dk è ö ÷t ¥ k0 ø ò d(k 0 -¥ æ dw i ( k0 + s ) ç x dk è + s ) A(k0 + s)e Wavepacket ö t÷ k0 ø 21 For those of you who want more: 1 0.5 0 k = k0 + s -0.5 Localized particle y -1 -2 j (x,t) = e æ dw ö -i ç w0 -k0 ÷t ¥ dk k0 ø è j (x,t) = e ò d ( k0 -¥ æ dw -iççw 0 -k0 dk è x 0 + s) A(k0 + s)e ö ÷÷ t ¥ k0 ø ò dk A(k)e 2 æ dw ö i ( k0 +s ) ç xt÷ è dk k0 ø æ dw ik çç xè dk ö t ÷÷ k0 ø -¥ Phase factor - goes to 1 in probability Wavepacket 22 For those of you who want more: 1 0.5 j (x,0) = ¥ ò dk A(k)e 0 ikx -0.5 -¥ Localized particle y -1 -2 ¥ j (x,t) = phase ò dk A(k)e 0 x æ dw ik çç xè dk 2 ö t ÷÷ k0 ø -¥ ¥ j (x,t) = phase ò dk A(k)e -¥ Wavepacket ik ( x-vgt ) = j (x - vgt,0) dw vg = dk 23 k = k0 Particular example of the Gaussian wavepacket. Wavepacket 24 j (x,t) @ ¥ ò dk e -k 2 4 a 2 ikx -iwt e e -¥ j (x,t) @ ¥ ò dk e é 1 ù - ê 2 +i t ú k2 2m û ë 4a eikx -¥ 1 - a 2 x 2 [1+it t ] j (x,t) @ e 1 + it t P(x,t) @ 1 1+ t t 2 2 e k2 w (k) = 2m Use same integral as before t= m 2 a2 - 2a 2 x 2 éë1+t 2 t 2 ùû Wavepacket 25 P(x,t) @ 1 1+ t t 2 2 e - 2a 2 x 2 éë1+t 2 t 2 ùû Wavepacket t= m 2 a2 26 P(x,t) @ t = 0t 1 1+ t t 2 2 e - 2a 2 x 2 éë1+t 2 t 2 ùû 1 - a 2 x 2 [1+it t ] j (x,t) @ e 1 + it t t = 2t Zero-momentum wavepacket: t = 5t Wavepacket Spreads but doesn’t travel! It has many positive k components as it has negative k components. 27 P(x,t) @ 1 1+ t2 t 2 e - 2a 2 x 2 éë1+t 2 t 2 ùû t is characteristic time for wavepacket to spread t= t for macroscopic things: m ≈ 10-3kg; 1/a ≈ 10-2 m t ≈ 1027 s ≈ 1020 yr !! long m 2 a 2 t for atomic scale: m ≈ 10-?kg; 1/a ≈ 10-? m t ≈ 10-? s t for nuclear scale: m ≈ 10-?kg; 1/a ≈ 10-? m t ≈ 10-? s Wavepacket 28 FREE PARTICLE QUANTUM WAVEPACKET - REVIEW • • • Gaussian superposition of free-particle eigenstates (of energy and momentum!) Localized in space means dispersed in momentum and vice versa. Look at time-dependent probability distribution: packet broadens and moves Wavepacket 29