Describe in words how you interpret this quantity. Precisely what information do you get from x?

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1 WAVE FUNCTIONS AND PROBABILITY 1 I: Thinking out the wve function In quntum mechnics, the term wve function usully refers to solution to the Schrödinger eqution, Ψ(x, t) i = 2 2 Ψ(x, t) + V (x)ψ(x, t), t 2m x 2 where V (x) is the potentil energy experienced y prticle of mss m nd Ψ(x, t) is the wve function in this one-dimensionl exmple. A. Let s sy you hve system where the wve function is of the form: Ψ 1 (x, t) = f(x)e iωt where f(x) is some rel-vlued function of x. 1. Is Ψ 1 (x, t) 2 rel? Is it positive? Do your nswers mke sense given the physicl mening (s discussed in clss) of Ψ 1 (x, t) 2? 2. Does Ψ 1 (x, t) depend on time? Does Ψ 1 (x, t) 2 depend on time? 3. Write down n expression for x. Does it depend on time? Is it rel? Descrie in words how you interpret this quntity. Precisely wht informtion do you get from x?. Write down n expression for g(x) where g(x) is ny rel-vlued function of x. Does it depend on time? Agin, how would you physiclly interpret g(x) (hint: think out wht you would ctully mesure)? PHYS 3220 Tutorils S. Goldher, S. Pollock, nd the Physics Eduction Group University of Colordo, Boulder

2 WAVE FUNCTIONS AND PROBABILITY 2 B. Now let s sy your system is it more complex (pun intended): Ψ 2 (x, t) = f(x)e iωt + g(x)e 2iωt where f(x) nd g(x) re rel functions of x which re orthogonl to ech other. 1. Is Ψ 2 (x, t) 2 rel? Is it positive? Do your nswers mke sense given the physicl mening of Ψ 2 (x, t) 2? 2. Does Ψ 2 (x, t) 2 depend on time? 3. Write down n expression for x. Does it depend on time? Descrie the difference(s) etween this result nd the result for section A.3 ove. Even though f nd g re unknown functions of x, do your est to give physicl description or interprettion of this new result for x for the stte Ψ 2. Check your results with tutoril instructor. PHYS 3220 Tutorils S. Goldher, S. Pollock, nd the Physics Eduction Group University of Colordo, Boulder

3 WAVE FUNCTIONS AND PROBABILITY 3 C. Now, we will del with new wve function t single moment in time, ψ 3 (x) = Ψ 3 (x, t = t 0 ), represented y the grph elow ( sine curve from π/ to 5π/ nd zero everywhere else). Ψ 3 x A 1. Find vlue of A which will normlize ψ 3 (x). x 2. Using physicl rguments (i.e., without doing the integrl), wht do you think x is? (If you feel uncertin, you cn check y doing the integrl) 3. We wnt to find the stndrd devition for x for this system. First, do you think tht x 2 is lrger/the sme/smller (circle one) thn x 2? Now, ctully clculte x 2.. Wht is σ 2 x? Wht is the proility tht you will find the prticle represented y ψ 3 (x) in the rnge x ± σ x? (Recll tht σ 2 x x 2 x 2 ). PHYS 3220 Tutorils S. Goldher, S. Pollock, nd the Physics Eduction Group University of Colordo, Boulder

4 WAVE FUNCTIONS AND PROBABILITY D. Now we somehow crete system where for n instnt, the wve function, ψ (x) = Ψ (x, t = t 0 ), looks like the grph elow. Ψ x A 1. Find the vlue of A which will normlize ψ (x). x 2. Using physicl rguments (i.e., without doing the integrl), wht do you think x is? (If you feel uncertin, you cn check y doing the integrl) 3. Estimte x 2 nd σ x. Indicte on the grph ove the rnge which you think represents x ± σ x. Bonus (i.e., come ck to this if you hve time fter finishing the rest of the tutoril), clculte x 2 nd σ 2 x.. How do you physiclly interpret σ x? 5. Wht re the possile vlues of mesurement of x on ny of these identicl systems? Do you expect to mesure x equl to the expecttion vlue of x? Check your results with tutoril instructor. PHYS 3220 Tutorils S. Goldher, S. Pollock, nd the Physics Eduction Group University of Colordo, Boulder

5 II: Clssicl current WAVE FUNCTIONS AND PROBABILITY 5 A. Consider thin, insulted wire with current which depends on the position long the wire. Let the current e given s I(x), where positive vlue of I represents current flowing to the right. I x Student A defines Q (t) to e the totl electric chrge in the wire etween points nd (see figure ove). Student B points out tht since chrge cnnot e creted or destroyed (i.e., chrge is conserved), Q cnnot e function of time. You re clled in to settle the dispute. Could Q depend on time? Wht is your resoning? B. No mtter wht you sid ove, suppose we told you we hd set up sitution where t n instnt of time, t 0, we hd mesured I() > I(). 1. Wht does this sitution imply out the time dependence of Q? 2. Construct formul for the time derivtive of Q in terms of I() nd I() PHYS 3220 Tutorils S. Goldher, S. Pollock, nd the Physics Eduction Group University of Colordo, Boulder

6 WAVE FUNCTIONS AND PROBABILITY 6 Useful Formuls x sin 2 (x x 0 )dx = x 2 sin 2 (x x 0 )dx = 1 2 sin 2 (x x 0 )dx = ( x 2 x 2 0 ( x x0 2 sin(2(x x ) 0)) cos(2(x x 0)) 8 x sin(2(x x 0)) ) ( x 3 6x cos(2(x x 0 )) + (3 6x 2 ) sin(2(x x 0 )) ) Useful Formuls x sin 2 (x x 0 )dx = x 2 sin 2 (x x 0 )dx = 1 2 sin 2 (x x 0 )dx = ( x 2 x 2 0 ( x x0 2 sin(2(x x ) 0)) cos(2(x x 0)) 8 x sin(2(x x 0)) ) ( x 3 6x cos(2(x x 0 )) + (3 6x 2 ) sin(2(x x 0 )) ) Useful Formuls x sin 2 (x x 0 )dx = x 2 sin 2 (x x 0 )dx = 1 2 sin 2 (x x 0 )dx = ( x 2 x 2 0 ( x x0 2 sin(2(x x ) 0)) cos(2(x x 0)) 8 x sin(2(x x 0)) ) ( x 3 6x cos(2(x x 0 )) + (3 6x 2 ) sin(2(x x 0 )) ) PHYS 3220 Tutorils S. Goldher, S. Pollock, nd the Physics Eduction Group University of Colordo, Boulder