8.04 Spring 2013, Problem Set 2 Solution

Problem Set 2 8.04 Spring 2013 Solutions February 21, 2013 Problem 1. (10 points) Wave-riding Mechanics (a) (4 points) Given a dispersion relation ω(k), the phase velocity vp is defined as vp ≡ ω , k (1) and is the rate at which a single plane wave (i.e. a single k-mode) moves within a wavepacket. The group velocity vg , on the other hand, is defined as vg ≡ ∂ω , ∂k (2) and is the rate at which the envelope, or the “pattern” of a wavepacket moves1 . Note that both are in general functions of k. For gravity waves2 , the dispersion relation is given by ω= gk tanh(kd). The open ocean is deep, so kd ≈ ∞, and tanh(kd) ≈ 1. This means ω ≈ √ kg, and so g k vp ≈ vg ≈ (3) 1 2 (4a) g . k (4b) In deep water, the phase velocity is thus twice as quick as the group velocity. (b) (3 points) From the dispersion relation (Equation 3), we see that as waves approach the beach (i.e. as d decreases), the angular velocity ω decreases. √ Furthermore, as one approaches the beach d gets small, so tanh(kd) ≈ kd and ω ≈ k gd. This gives √ vg ≈ gd, which gets smaller as d gets smaller. Putting all this together, we see that the front of the envelope of the wavepacket slows down as the individual plane waves farther behind (i.e. still in deeper water) continue to move at a fairly quick rate. In other words, the difference between vf and vg changes. This causes the waves to “scrunch up” at the front of the envelope and the height of the waves to increase (since the total volume of water must remain constant). So the fact that waves appear to get taller as they approach the beach is not just an illusion. Eventually, the heights of the waves increase to such a point that nonlinearities set in, and the waves break. 1 Highly recommended — To get a feel for the distinction between phase and group velocities, check out http://galileoandeinstein.physics.virginia.edu/more_stuff/Applets/wavepacket/wavepacket.html 2 Not to be confused with gravitational waves, which are ripples in spacetime. 1 (c) (3 points) The surfer’s impression is accurate. Individual plane waves advance “through” the envelope of a wavepacket quickly at rate of the phase velocity, but then die down in amplitude as they approach the front of the envelope in order to maintain the en­ velope’s shape, which advances slowly at rate of the group velocity. To see an extreme case that illustrates this point, set the group velocity to zero in the applet cited in the footnote to part (a). One sees that at the nodes of the envelope, the individual plane waves die down completely. From the applet and the discussion above, we know that individual plane waves inter­ fere to reach their maximum amplitude in the mid-section of a wavepacket’s envelope. As the wavepacket advances, it retains the shape of its envelope, so an observer stand­ ing at a fixed location experiences alternating groups of “big” (large amplitude) waves and “small” (small amplitude) waves. 2 Problem Set 2 8.04 Spring 2013 Solutions February 21, 2013 Problem 2. (10 points) Why I Don’t Play Craps (a) (3 points) Since a fair die is equally likely to give s from 1 to 12, one intuitively expects to get (s) = 6.5. (5) This is correct, but it pays to set things up a little more formally so that we can practice some of the techniques that will be useful when doing quantum problems. The definition of (s) is (s) = s P(s). (6) s 3 For a fair die, P(s) is equal to 1/12 for s = 1, 2 . . . 12, so 1 (s) = 8 8 s= s=1 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 = 6.5. 12 (7) (b) (3 points) First we define Δs2 to be Δs2 ≡ ((s − (s))2 ), (8) √ and Δs to be Δs = Δs2 (i.e. Δs2 should be thought of as (Δs)2 and not Δ(s2 )). One way to tackle this problem would be to evaluate Δs2 by brute force: Δs2 = (s − (s))2 P(s). (9) s 4 However, we saw in lecture that Δs2 = (s2 ) − (s)2 . (11) 12 1 A more sophisticated way to write P(s) would be to say P(s) = 12 i=1 δ(s − i). Graphically, this would look like eight infinitely thin spikes centered on s = 1, 2 . . . 12, which is just a way of saying that any given throw of the die will give some integer from 1 to 12 and nothing in-between. (The expectation value, being like an average, is allowed to take on non-integer values). Using this sort of formalism, the expectation values are integrals, which more closely resembles the sorts of expectation values that we deal with in quantum a I ∞ e 12 I∞ 1 mechanics. For example, when finding (s), we have (s) ≡ −∞ s P(s)ds = 12 −∞ s i=1 δ(s − i) ds = eI a ∞ 12 12 1 1 s δ(s − i)ds = 12 i=1 i=1 i, which is the same as Equation 6. Note that our P(s) satisfies the 12 −∞ usual properties required for probability distribution functions, such as being positive everywhere and being I∞ properly normalized to one (i.e. −∞ P(s)ds = 1). 4 As a reminder: Δs2 ≡ ((s − (s))2 ) = (s2 − 2s(s) + (s)2 ) = (s2 ) − 2(s(s)) + ((s)2 ). If you’re not convinced that (a + b) = (a) + (b), try writing out the expectation value integral explicitly. Now, unlike s, there is nothing random about (s) — it’s simply a number. This means that any (s)’s coming sailing out of the (. . . )’s, so that (s(s)) = (s)(s) = (s)2 and ((s)2 ) = (s)2 , giving p Δs2 = (s2 ) − 2(s)2 + (s)2 = (s2 ) − (s)2 ⇒ Δs = (s2 ) − (s)2 . (10) 3 3 Thus, all that remains is for us to work out (s2 ), which we can do in a similar way to the manipulations in Equation 7: 12 (s2 ) = 1 X 2 325 s = . 12 s=1 6 (12) Putting everything together gives Δs = 3.45. (13) (c) (4 points) Let st represent the total number of spots shown by the two dice, and let s1 and s2 be the number of spots shown by the first and second die respectively. By definition, st = s1 + s2 , so (st ) = (s1 + s2 ) = (s1 ) + (s2 ) = 6.5 + 6.5 = 13. (14) Computing Δst is a little trickier. To avoid writing square root signs, let’s consider Δs2t . We start by using the shortcut we found in Equation 10: Δs2t = (st2 ) − (st )2 = ((s1 + s2 )2 ) − ((s1 ) + (s2 ))2 = (s21 + 2s1 s2 + s22 ) − (s1 )2 − 2(s1 )(s2 ) − (s2 )2 = (s21 ) + 2(s1 s2 ) + (s22 ) − (s1 )2 − 2(s1 )(s2 ) − (s2 )2 . (15) Now, because the two die throws are independent, the joint probability P(s1 , s2 ) of obtaining s1 for the first die and s2 for the second die is P(s1 )P(s2 )5 . we have � �� � X X X X (s1 s2 ) = s1 s2 P(s1 , s2 ) = s1 s2 P(s1 )P(s2 ) = s1 P(s1 ) s2 P(s2 ) = (s1 )(s2 ), s1 ,s2 s1 ,s2 s1 s2 (16) which means our expression for Δs2t reduces to Δs2t = (s12 ) − (s1 )2 + (s21 ) − (s2 )2 = 2Δs2 , (17) where Δs refers to the spread defined for the single die throw analyzed in part (b). This gives √ Δst = 2Δs2 = 4.88. (18) 5 Please excuse the sloppy notation — P(s1 , s2 ) is of course not the same function as P(s). After all, P(s1 , s2 ) has two inputs while P(s) only has one. 4 Commentary: If we repeated this problem for N dice, we would have found (st ) = N (s) √ Δst = N Δs2 , (19a) (19b) which means the fractional spread takes the form Δst 1 Δs =√ . (st ) N (s) (20) Now, suppose we forget about the dice and think about s as the result of some experi­ mental measurement. Because of experimental uncertainties, the measurement comes with some error Δs, and Δs/(s) is the percentage error on the measurement. What Equation 20 tells us is that if we repeat the measurement√over and over again and average our results, the percentage error will go down as 1/ N . Note that this behavior depends crucially on Equation 16, which was a result of each die throw/measurement being independent. Intuitively, what’s happening is that with independent random errors, each individual measurement may be too high or too low, but on average the measurements will be high half the time and low the rest of the time. Thus, as more and more measurements are taken, the errors average down. On the other hand, if the measurements are off because of some systematic bias (e.g. if the measurements are always too high because of some instrumental miscalibration) then the errors do not average down. This is why physicists work so hard to eliminate sources of systematic error in their experiments. 5 Problem Set 2 8.04 Spring 2013 Solutions February 21, 2013 Problem 3. (5 points) Dimensions of ψ (a) (2 points) To determine the dimension of ψ we start with Born interpretation: dP (x) = ψ (x) |2 dx , (21) i.e. the probability to find the particle in an infinitesimal interval dx around a position x is equal with the probability density |ψ (x) |2 times the infinitesimal interval. Since a probability is a dimensionless quantity, [P] = 1, we have: 1 = [ψ (x)]2 × L ⇒ 1 [ψ (x)] = √ L (22) (b) (3 points) As seen in lecture and proved below, we have: 2 dP (k) = ψ̃ (k) dk. Recalling that k was defined as k = reasoning as above to get, 2π , λ 2 1 = ψ̃ (k) × (23) we see that [k] = L1 , so we can follow the same 1 L ⇒ 6 ψ˜ (k) = √ L. (24) Problem Set 2 8.04 Spring 2013 Solutions February 21, 2013 Problem 4. (15 points) Fourier Transforms and Expectation Values (a) (5 points) By definition of the expectation value of an operator, we have ∞ ψ ∗ (x) p̂ ψ(x) dx, (p) = (25) −∞ where the hat on p̂ emphasizes the fact that p̂ is an operator and not a number. An immediate consequence of this is that except for in certain special cases, we can’t swap the order of the terms in Equation 25, i.e. ψ ∗ (x) pˆ ψ(x) = 6 pˆ ψ ∗ (x) ψ(x). We can see this explicitly for this particular problem by plugging in the form of the momentum operator: ∞ n ∂ (p) = ψ ∗ (x) ψ(x) dx. (26) i ∂x −∞ We can see that the derivative acts only on the second copy of ψ, whereas if we had written p̂ ψ ∗ (x) ψ(x) it would’ve acted on the product of ψ(x) and ψ ∗ (x), which we know gives a different result from the product rule in calculus. Let us now proceed by substituting into Equation 26 the definition of the Fourier transform: ∞ (p) = −∞ 1 2π 1 = 2π 1 = 2π ∞ = −∞ ∞ −∞ ∞ −∞ ∗ ∞ 1 ˜ √ eikx ψ(k)dk 2π −∞ −∞ ∞ ∞ n ∂ ˜ eikx ψ(k)dk dx e−iqx ψ˜∗ (q)dq i ∂x −∞ −∞ ∞ ∞ −iqx n ∂ ˜ eikx dx dq dk ψ̃ ∗ (q)ψ(k)e i ∂x −∞ −∞ ∞ ∞ n ∂ ikx ψ˜∗ (q)ψ̃(k) e−iqx e dx dq dk. i ∂x −∞ −∞ 1 √ 2π ∞ iqx e ˜ ψ(q)dq n ∂ i ∂x dx (27a) (27b) (27c) (27d) A lot happened in the last few lines, and here are some subtleties to be aware of: • In Equation 26, there are two copies of ψ(x). When substituting these for their Fourier space representations, it is crucially important to use different variables for the Fourier variables (i.e. to not use k for both of them). To see this, recall that the integrals are really just fancy summations with q or k as dummy summation variables, and consider the following example, P2 which serves as an analogy. PSuppose we’re trying to find the product of a ≡ i=1 i = 1 + 2 = 3 and b ≡ 2j =1 j 2 = 1 + 4 = 5. The answer is of course 15, but let’s do this formally: � 2 � � !� 2 ! 2 X 2 X X X 2 ab = i j = ij 2 = 1 + 4 + 2 + 8 = 15. (28) i=1 j=1 i=1 j=1 7 If we (incorrectly) use the same dummy index for the two summations, it is easy to forget that we’re doing two sums, which leads to missing many of the terms in the sum: � 2 ! �� 2 ! � 2 X X X 2 ab = i i → ii2 = 1 + 8 = 9 (Wrong!) (29) i=1 i=1 i=1 where the right arrow signifies an incorrect logical step. • In going from Equation 27a to 27b, we brought the momentum operator and the complex conjugate inside the Fourier integrals. This is allowed because differen­ tiation and taking the complex conjugate of something are both linear operations acting on integrals, which are just sums. By definition, linear operations are ones where the same answer is obtained regardless of whether we sum (i.e. integrate) first and then “operate” or “operate” first and then sum. • In going from Equation 27b to 27c, we switched the order of integration. This is allowed by Fubini’s theorem (a purely mathematical result) thanks to the inde­ pendence of our three integration variables x, q, and k. ˜ • In going from Equation 27c to 27d, ψ(k) passed right through the derivative, because it is a function of k and not of x. Proceeding with the algebra from Equation 27d, we have Z ∞  Z ∞Z ∞ 1 ∗ −iqx ikx (p) = ψ˜ (q)ψ̃(k) e nk e dx dq dk 2π −∞ −∞ −∞ Z ∞  Z ∞Z ∞ 1 ∗ i(k−q)x e dx dq dk = ψ˜ (q)ψ̃(k) nk 2π −∞ −∞ −∞ " , . (30a) (30b) =2πδ(k−q) Z ∞ Z ∞ ψ̃ ∗ (q)ψ̃(k) nk δ(k − q)dq dk = Z−∞ ∞ = Z−∞ ∞ = (30c) −∞ ψ̃ ∗ (k) nk ψ̃(k) dk (30d) |ψ̃(k)|2 nk dk, (30e) −∞ which is our desired result. Note that Equation 30d is very similar in form to Equation 25, except we have k instead of x and ψ˜ instead of ψ. This suggests that while the momentum operator p̂ takes the form of a derivative in real coordinate space, in Fourier space it is simply a multiplicative operator. (b) (5 points) The steps to follow are the same as in part (a), except for the operator p̂ 8 which must be applied twice: 2 p̂ = = = = = = Z ∞ Z ∞   ∂ ∂ dxψ (x) −in −in ψ(x) (31a) ∂x ∂x −∞ Z ∞ ∂2 2 dxψ ∗ (x) 2 ψ(x) (31b) −n ∂x −∞     Z ∞ Z ∞ Z ∞ 2 1 1 2 iqx ∗ ∂ ikx √ dx √ dqe ψ̃(q) dke ψ̃(k) (31c) −n ∂x2 2π −∞ 2π −∞ −∞  Z ∞  2   Z ∞ Z ∂ ikx n2 ∞ −iqx ∗ dx dqe dk e ψ̃ (q) ψ̃(k) (31d) − 2π −∞ ∂x2 −∞ −∞ Z ∞  Z ∞  Z n2 ∞ −iqx ∗ 2 ikx dx dqe dk −k e ψ̃(k) ψ̃ (q) (31e) − 2π −∞ −∞ −∞ Z ∞ Z ∞ Z ∞ 1 2 ∗ 2 dqψ̃ (q) dkk ψ̃(k) dx ei(k−q)x (31f) n 2π −∞ −∞ −∞ " {z , . ∗  δ(k−q) = dk|ψ̃(k)|2 (nk)2 . (31g) −∞ Again we see that the operator p̂ becomes a multiplicative operator in the momentum space. (c) (5 points) We wish to prove that Z ∞ |ψ̃(k)|2 f (nk)dk. (f (p̂)) = (32) −∞ The left hand side can be Taylor expanded to give p̂2 "" f (0) + . . . ) (33a) 2! (p̂2 ) "" f (0) + . . . (33b) = f (0) + (p̂)f " (0) + 2! To evaluate this, we need to know how to work out expectation values of powers of ∂ and p̂, i.e. (p̂n ). One way to do this would be to go all the way back to p̂ = ni ∂x n to find (p̂ ) by brute force, but an easier way would be to use our result from the previous part. There, we showed that in Fourier space, the momentum operator takes the simple form of multiplication by nk. It follows, then, that powers of p̂ correspond to powers of nk in Fourier space, which means Z ∞ Z ∞ 1 "" " 2 |ψ̃(k)| nk dk + f (0) |ψ̃(k)|2 (nk)2 dk + . . . (34a) (f (p̂)) = f (0) + f (0) 2! −∞ −∞  Z ∞ 2 (nk) 2 " "" = |ψ̃(k)| f (0) + nkf (0) + f (0) + . . . dk (34b) 2! −∞ Z ∞ = |ψ̃(k)|2 f (nk) dk. (34c) (f (p̂)) = (f (0) + p̂f " (0) + −∞ 9 This is the same I ∞as Equation 32, so our proof is complete. Note that we implicitly used the fact that −∞ |ψ̃(k)|2 dk = 1. This can be seen either as a mathematical statement (Parseval’s theorem from Fourier analysis), or a physical one (probabilities need to add up to 1 whether we work in real space or Fourier space). 10 Problem Set 2 8.04 Spring 2013 Solutions February 21, 2013 Problem 5. (15 points) Delta Functions (a) i. (1 point) Z 1 δ(x + 2) (x3 − 3x2 + 2x − 1) dx = (−2)3 − 3(−2)2 + 2(−2) − 1 = −25. (35) −3 ii. (1 point) Z ∞ δ(x − π) (cos(3x) + 2) dx = cos(3π) + 2 = 1. (36) 0 iii. (1 points) Z 1 δ(x − 2) e|x|+3 dx = 0, (37) −1 because x = 2 is not in the range [−1, 1]. (b) i. (1 point) Z ∞ ∞ Z x δ(x)f (x) dx = 0f (0) = 0 = −∞ ⇒ 0f (x) dx xδ(x) = 0. (38) δ(−x) = δ(x). (39) −∞ ii. (1 point) Z ∞ Z δ(−x)f (x) dx = f (0) = −∞ ∞ δ(x)f (x) dx ⇒ −∞ iii. (1 point) Assume c > 0, and let u ≡ cx. Then Z ∞ Z ∞ Z ∞ e u a du 1 δ(x) δ(cx)f (x) dx = δ(u)f = f (0) = f (x) dx. c c c −∞ −∞ −∞ c (40) If c < 0, then the limits of integration get swapped during the substitution, so Z ∞ Z ∞ Z ∞ e u a du 1 δ(x) δ(cx)f (x) dx = − δ(u)f = − f (0) = f (x) dx. (41) c c c −∞ −∞ −∞ −c Putting these two results together gives δ(cx) = 1 δx. |c| iv. (1 point) This can be shown directly simply by letting f (x) ≡ δ(x): Z ∞ δ(a − x)f (x − b) dx = f (a − b) = δ(a − b). −∞ 11 (42) (43) v. (1 point) Z ∞ ∞ Z δ(x − a)f (x)g(x) dx = f (a)g(a) = −∞ δ(x − a)f (a)g(x) dx (44a) −∞ ⇒ f (x)δ(x − a) = f (a)δ(x − a). (44b) (c) For each of the proposed forms of the delta function, we essentially need to show that Z ∞ f (y)δ(y − x) dy = f (x). (45) −∞ i. (4 points) Consider the inverse Fourier transform equation: Z ∞ 1 f (x) = √ eikx f˜(k)dk. 2π −∞ (46) Now suppose we substitute into this the formula for the Fourier transform, that is, Z ∞ 1 ˜ f (k) = √ (47) e−ikx f (x)dx. 2π −∞ The result is  Z ∞  Z ∞ 1 ikx −iky f (x) = e e f (y)dy dk. (48) 2π −∞ −∞ (Note the crucial renaming of the dummy variable in Equation 47 from x to y. See the first bullet point in the Problem 3 solutions for more details). Changing the order of integration gives  Z ∞  Z ∞ 1 ik(x−y) f (x) = f (y) e dk dy, (49) 2π −∞ −∞ " , . δ(x−y) where we identified the term in the parentheses as a delta function by comparing our expression to Equation 45 (note that δ(x−y) = δ(y −x) from Part b ii above). We can thus conclude that Z ∞ 1 δ(x) = eikx dk. (50) 2π −∞ ii. (4 points) We want to show that   Z ∞ 1 −x2 /a2 f (x) lim √ e dx = f (0). a→0 a π −∞ To do so, we first take the limit outside the integral: Z ∞ 1 2 2 lim f (x) √ e−x /a dx. a→0 −∞ a π 12 (51) (52) We now non-dimensionalize by letting u = x/a: Z ∞ 1 2 lim f (ua) √ e−u du. a→0 −∞ π (53) The only remaining factor of a resides in f (ua), so we can simply take the limit and say f (ua) → f (0), which then comes out of the integral because it is now a constant: Z ∞ 1 2 √ e−u du = f (0), f (0) lim (54) a→0 −∞ π where we have recognized that the integral is a standard Gaussian integral that evaluates to 1. We have thus shown that Equation 51 is true and δ(x) = 2 2 lima→0 a√1 π e−x /a . 13 Problem Set 2 8.04 Spring 2013 Solutions February 21, 2013 Problem 6. (30 points) Qualitative Structure of Wavefunctions (a) (8 points) Because these are the official electronic solutions, we have chosen to plot these wavefunctions rather than sketch them. It is worth noting, however, that being able to sketch a function in a qualitatively accurate way is an important skill, so definitely get some practice if you’re uncomfortable with it. (Remember, you won’t get to use your computer on exams!). In the plots below, we sketch the real part of the wavefunction ψ(x) in blue and the corresponding probability distribution |ψ(x)|2 = ψ ∗ (x)ψ(x) in red. • ψ1 (x) = δ(x − 1). Delta functions are infinitely narrow and infinitely tall, so they can’t really be plotted. Shown below is a Gaussian approximation (do Problem 6 c to see why this is sensible!). 20 15 10 5 !3 !2 !1 0 1 2 3 0 1 2 3 • ψ2 (x) = δ(x − 2). 20 15 10 5 !3 !2 !1 14 • Real part of the wavefunction: Re(ψ3 (x)) = Re(eix ) = Re(cos x + i sin x) = cos x. Probability distribution: |ψ3 (x)|2 = ψ3∗ (x)ψ3 (x) = e−ix eix = 1. 2 1 !10 !5 5 10 !1 !2 • Real part of the wavefunction: Re(ψ4 (x)) = Re(ei2x ) = Re(cos 2x + i sin 2x) = cos 2x. Probability distribution: |ψ4 (x)|2 = ψ4∗ (x)ψ4 (x) = e−i2x ei2x = 1 2 1 !10 !5 5 10 !1 !2 • ψ5 (x) = δ(x − 1) + δ(x − 2). 20 15 10 5 !3 !2 !1 0 15 1 2 3 • Real part of the wavefunction: Re(ψ6 (x)) = Re(ei2x + eix ) = Re(cos 2x + i sin 2x + cos x + i sin x) = cos 2x + cos x. Probability distribution: |ψ4 (x)|2 = ψ4∗ (x)ψ4 (x) = (e−i2x + e−ix )(ei2x + eix ) = 2 + eix + e−ix = 2 + 2 cos x 5 4 3 2 1 !10 !5 5 10 !1 !2 • ψ7 (x) = N if − a2 ≤ x ≤ a2 . The x-axis is plotted in units of a, while the y-axis is plotted in units of 0.1N . 120 100 80 60 40 20 !2 !1 1 2 (x−x0 )2 • ψ8 (x) = N e− a2 eik0 x . The x-axis is plotted in units of a, while the y-axis is plotted in units of 0.1N , x0 = 1 and k0 = 2. 1.0 0.5 -2 -1 1 -0.5 -1.0 16 2 (b) (4 points) In each case we must find ψ̃(k) ≡ • ψ˜1 (k) = √1 2π • ψ˜2 (k) = √1 2π • • • • • • I∞ −∞ δ(x − 1)e−ikx dx = √1 2π I∞ −∞ ψ(x)e−ikx dx. √1 e−ik . 2π I∞ δ(x − 2)e−ikx dx = √12π e−i2k . √ I∞ I∞ ψ˜3 (k) = √12π −∞ eix e−ikx dx = √12π −∞ ei(1−k)x dx = 2πδ(k − 1). (See Problem 6 if you’re not sure where the last equality came from). √ I∞ I∞ ψ˜4 (k) = √12π −∞ ei2x e−ikx dx = √12π −∞ ei(2−k)x dx = 2πδ(k − 2). I∞ ψ˜5 (k) = √12π −∞ (δ(x − 1) + δ(x − 2))e−ikx dx = √12π (e−ik + e−i2k ). I∞ I∞ ψ̃6 (k) = √12π −∞ (eix + ei2x )e−ikx dx = √12π −∞ ei(1−k)x + ei(2−k)x dx √ = 2π(δ(k − 1) + δ(k − 2)). I a/2 ψ˜7 (k) = √N2π −a/2 e−ikx dx = N π2 sin(ka/2) . k I∞ 2 2 2 2 √a e−i(k−k0 )x0 e−(k−k0 ) a /4 . ψ˜8 (k) = √N2π −∞ e−(x−x0 ) /a eik0 x e−ikx dx = N 2 −∞ Note that the operation of taking a Fourier transform is a linear one, so another way to find ψ5 and ψ6 would’ve been to compute ψ1 + ψ2 and ψ3 + ψ4 respectively. (c) (4 points) Again, the real part of the wavefunction is in blue and the probability distribution is in red: • ψ˜1 (k) = √1 e−ik , 2π so the real part is Probability distribution: |ψ̃1 (k)|2 = √1 cos k. 2π ψ˜1∗ (k)ψ̃1 (k) = 1 ik −ik e e 2π = 1 . 2π 1.0 0.5 !10 !5 5 10 !0.5 !1.0 At large k, P(k) ∼ divergent. 1 . 2π From the formula obtained in Problem 4, this makes (p2 ) 17 • ψ˜2 (k) = √1 e−i2k , 2π so the real part is Probability distribution: |ψ̃2 (k)|2 = √1 cos 2k. 2π 1 i2k −i2k ∗ ˜ ψ2 (k)ψ̃2 (k) = 2π e e = 1 . 2π 1.0 0.5 !10 !5 5 10 !0.5 !1.0 Also in this case, since P(k) tends to a constant at large k, the integral for (p2 ) is divergent. √ • ψ˜3 (k) = 2πδ(k − 1). 20 15 10 5 !3 !2 !1 0 1 2 3 Here P(k) = 0 at k = 1. In this case the uncertainty on p is zero. 18 • ψ˜4 (k) = √ 2πδ(k − 2). 20 15 10 5 !3 !2 !1 0 1 2 3 As in the previous case, P(k) = 0 at k 6= 2, and the uncertainty on p is zero. • ψ˜5 (k) = √12π (e−ik + e−i2k ), so the real part is √12π (cos k + cos 2k). Probability distribution: |ψ̃5 (k)|2 = ψ̃5∗ (k)ψ̃5 (k) = 1 (2 + eik + e−ik ) = π1 (1 + cos k). 2π 1 (e−i2k 2π 1.0 0.5 !10 !5 5 10 !0.5 !1.0 The asymptotic behavior of P(k) makes (p2 ) divergent. √ • ψ˜6 (k) = 2π(δ(k − 1) + δ(k − 2)). 20 15 10 5 !3 !2 !1 0 19 1 2 3 + e−ik )(ei2k + eik ) = Since P(k) = 0 for k = 6 1, 2, (p2 ) is finite, although, contrary to ψ3 , ψ4 , the uncertainty on p is nonzero. • ψ˜7 (k) = N 2 sin(ka/2) . π k 1.0 0.5 !20 !10 10 20 !0.5 !1.0 2 P(k) = N π2 sin (ka/2) , so, for large k, P(k) goes essentially like 1/k 2 , which tends k2 to zero too slowly to obtain a finite value of (p2 ): Z ∞ Z ∞ 2 2 (p ) = dk(nk) P(k) ∼ dk sin2 (ka/2) = ∞. −∞ −∞ 2 2 √a e−i(k−k0 )x0 e−(k−k0 ) a /4 . The x-axis is in units of ka/2, while the y-axis • ψ̃8 (k) = N 2 is in units of aN , x0 = 1 and k0 = 2. 1.0 0.5 -4 -2 2 4 -0.5 -1.0 In this case, P(k) ∼ e−2(k−k0 ) value of (p2 ). 2 a2 /4 , and it tends to zero fast enough to get a finite (d) (6 points) In general, one should look for maxima in |ψ(x)|2 to decide where it will most likely be found. The expectation value (x) can be thought of as an “average” position. With momentum, one repeats the same procedure but using |ψ̃(k)|2 instead. The narrower the peaks of these probability distribution functions, the more confident we can be that the particle’s position and momenta are measured to be near their “most likely values”. 20 • Particle #1: – Will certainly be found at x = 1, because |ψ(x)|2 = 0 everywhere else. – Momentum equally likely to take on any value, because |ψ̃(k)|2 is constant. • Particle #2: – Will certainly be found at x = 2, because |ψ(x)|2 = 0 everywhere else. – Momentum equally likely to take on any value, because |ψ̃(k)|2 is constant. • Particle #3: – Particle equally likely to be found anywhere, because |ψ(x)|2 is constant. – Momentum will certainly be nk = n, because |ψ̃(k)|2 = 0 for all k 6= 1. • Particle #4: – Particle equally likely to be found anywhere, because |ψ(x)|2 is constant. – Momentum will certainly be nk = 2n, because |ψ̃(k)|2 = 0 for all k 6= 2. • Particle #5: – Particle will either be at x = 1 or x = 2, with equal probability – Momentum most likely p = 2nπn, where n is any integer. • Particle #6: – Particle most likely to be found at x = 2nπ, where n is any integer. – Momentum will be either nk = n or nk = 2n with equal probability. • Particle #7: – Particle equally likely to be found anywhere between x = −a/2 and x = a/2. – Momentum will most likely be zero, but there is considerable width to this probability maximum, so measuring a value significantly different from zero would not be surprising. • Particle #8: – Particle most likely to be found at x = x0 , but peak is wide, so a spread in measured position would not be surprising. – Momentum will most likely be at nk = nk0 , but peak is wide, so a spread in measured momentum would not be surprising. (e) (2 points) We require the total probability of finding a particle somewhere to be 1, so properly normalized wavefunctions must satisfy Z ∞ 1= |ψ(x)|2 dx (55) −∞ For ψ8 (x), this means Z ∞ 2 |ψ(x)| dx = N 1= −∞ 2 Z ∞ −2(x−x0 )2 /a2 e −∞ 21 r π dx = N a 2 2  ⇒ N= 2 πa2  14 . (56) (f) (4 points) As noted above using symmetry, (x) = x0 and (p) = nk0 for ψ8 (x). If one wishes to be formal, one can compute the integrals explicitly. For example, (x) for ψ8 (x) is given by r Z ∞ Z ∞ 2 2 2 ∗ (x) = ψ8 (x)xψ8 (x) dx = e−2(x−x0 ) /a xdx = x0 . (57) 2 πa −∞ −∞ Z ∞ Z ∞ a 2 2 ∗ (p) = ψ̃8 (k)nkψ̃8 (k) dk = √ e−(k−k0 ) a /2 nkdk = nk0 . (58) 2π −∞ −∞ p p The uncertainties are given by Δx ≡ (x2 ) − (x)2 and Δp ≡ (p2 ) − (p)2 , so our next step is to find (x2 ) and (p2 ). For ψ8 we have Z 2 r ∞ (x ) = ψ8∗ (x)x2 ψ8 (x) dx = Z−∞ ∞ 2 πa2 Z ∞ x2 e− 2(x−x0 )2 a2 −∞ ψ̃8∗ (k)n2 k 2 ψ̃8 (k) dk = −∞ Z ∞ (k−k0 )2 a2 n2 a = √ n2 k 2 e− 2 dk = 2 + (nk0 )2 , a 2π −∞ (p2 ) = dx = a2 + x02 4 (59a) (59b) so Δx = a/2 and Δp = n/a. This means Δx · Δp = n/2, so the Gaussian wavefunction saturates the Uncertainty Principle i.e. it is a minimum uncertainty wavepacket. (g) (2 points) The position space wavefunction ψ8 (x) gets narrower and taller as a → 0. We thus expect Δx to tend to zero as this happens, which is confirmed by what we found in part (f), where Δx ∝ a. Conversely, the momentum space wavefunctions ψ̃(k) get broader and flatter, and Δp tends to infinity. This is again confirmed by the fact that with ψ8 , we have Δp ∝ 1/a. 22 Problem Set 2 8.04 Spring 2013 Solutions February 21, 2013 Problem 7. (15 points) Why the Wavefunction should be Continuous (a) (7 points) The process here is exactly the same as what we did for parts (e) and (f) of the Problem 6. First we find N : Z ∞ Z a/2 1 2 2 dx = N 2 a ⇒ N = √ . (60) 1= |ψ7 (x)| dx = N a −∞ −a/2 Now we find (x2 ) because we need it to find Δx: Z 2 ∞ 1 = a ψ7∗ (x)x2 ψ7 (x) dx (x ) = −∞ Z a/2 x2 dx = −a/2 a2 . 12 (61a) Since (x) = 0 byp symmetry (work it out explicitly if you’re skeptical!), we have Δx ≡ p (x2 ) − (x)2 = (x2 ), and a Δx = √ . (62) 2 3 The spread Δx is thus smaller than the value we found for ψ8 in the previous problem. (b) (8 points) We found in Problem 6 that N ψ̃7 (k) = √ 2π Z r a/2 −ikx e dx = N −a/2 2 sin(ka/2) = π k By symmetry, (p) = 0. As for (p2 ), we have Z ∞ Z 2 ∗ 2 2˜ (p ) = ψ̃7 (k)n k ψ7 (k) dk = −∞ ∞ −∞ 2n2 sin2 πa r  2 sin(ka/2) . πa k ka 2 (63)  dk = ∞. (64) To have a finite (p2 ), we need the integral in its corresponding formula to be convergent. Since the integrand is a nonnegative function, we must require that, for large k, k 2 |ψ̃(k)|2 ≤ C2 , k 1+E for some constants C and t. This implies, for large k, |ψ̃(k)| ≤ 23 C k 3+E 2 . Problem Set 2 8.04 Spring 2013 Solutions February 21, 2013 Problem 8. (Optional) Smooth Wavefunctions give finite expectation values (a) The following picture shows the plots of ψ7b (x) for b = 1, 1/2, 1/4, 1/8, 1/16, and a = N = 1. Note how, as b → 0, the wavefunction approaches ψ7 (x). From the picture we note that, for b << a, the area between the graph of ψ7b (x)/N 1.0 0.8 0.6 0.4 0.2 -2 -1 1 2 and the x-axis is approximately given by a rectangle centered at the origin of height 1 and width a.√The same goes for the area between (ψ7b (x)/N )2 and the x − axis, and thus N ∼ 1/ a. (b) As can be found with Mathematica, or consulting a table of Fourier transforms, we know that the Fourier transform of tanh(x) is, up to a divergent constant r   π kπ i csch . (65) 2 2 Because the Fourier transform is linear, and since we will subtract two tanh’s, the constants we are neglecting will cancel, so we can ignore them. From (65) we now obtain the Fourier transform of tanh(Ax + B). Suppose f˜(k) is the Fourier transform of f (x). The Fourier transform of f (Ax) is then Z Z y 1 1 −ikx √ √ dxe f (Ax) = dye−ik A f (y) = 2π 2π|A|   Z k 1 1 1 ˜ k −i A y √ f (y) = , = dye f |A| 2π |A| A where we made the change of variable y = Ax. The Fourier transform of f (x + B) is Z Z 1 1 −ikx √ dxe f (x + B) = √ dye−ik(y−B) f (y) = 2π 2π 24 Z 1 =e √ dye−iky f (y) = eikB f˜(k), 2π where we made the change of variable ye = xa+ B. Putting together these results, we x+ a find that the Fourier transform of tanh b 2 is ikB ie ika 2 r   π kbπ |b| csch . 2 2 Using the linearity of the Fourier transform, we finally obtain r r    −ika ika N π kbπ π kbπ ˜ ψ7b (k) = ie 2 |b| csch − ie 2 |b| csch 2 2 2 2 2 r π |b| sin ka 2 =N , 2 sinh kbπ 2 = and since b > 0, this is the result we were expected to find. (c) We have that, for large k, |ψ̃7b (k)|2 = N 2 ) π b2 sin2 ( ka 2 , |k|πb 2 e which means that |ψ̃7b (k)|2 dies exponentially at infinity, whereas we saw that |ψ̃7 (k)|2 is precisely equal to zero out of a bounded domain. From the considerations made in Problem 7 (b), we know that (p̂2 ) is certainly finite. (d) By symmetry, (p̂) = 0. Thus Δp2 = (p̂2 ), and we have 2 Δp ∝ Z = Z dkk 2 ) 1 b2 sin2 ( ka 2 2 πkb = a sinh ( 2 ) Z ) dy e y a2 1 b2 sin2 ( ya dy e y a2 1 b2 /2 1 2b ≈ , 2 πy 2 πy ∝ b b a sinh ( 2 ) b b a sinh ( 2 ) ab where we made the change of variable y = kb. Thus we conclude that 1 Δp ∝ √ . ab 25