Do the one-dimensional kinetic energy and momentum operators commute? If not, what operator does their commutator represent?
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1 1 Problem 1 Do the one-dimensionl kinetic energy nd momentum opertors commute? If not, wht opertor does their commuttor represent? KE ˆ h m d ˆP i h d 1.1 Solution This question requires clculting the commuttor of the opertors given. [ ] KE, ˆ ˆP [ h d m i h d i h d h m i [ ] h3 d d d d m i [ ] h3 d 3 d 3 m d ] The opertors commute; ll done. Problem Given the following wvefunction, describing some quntum prticle, nded in bsis of eigenfunctions of loction opertor, Ψ C left ψ left + C right ψ right + C top ψ top + C bottom ψ bottom NOTE:the eigenfunction ψ left is ssocited with n eigenvlue of LEFT, nd so on for the other eigenfunctions. Wht is the probbility of observing vlue of TOP before ny licit mesurement is mde. If we hve detector tht mesures the LEFT loction, nd signl is detected there fter prticle is projected towrds the rry of detectors:. How would you write the ression for the wvefunction? b. Wht is the probbility of observing vlue of LEFT? 1
2 c. If mesurement ws mde on the wvefunction 5 hours lter, wht would be the probbility of mesuring vlue of BOTTOM ssume no interctions of the prticle with externl fields, prticles, or other hve occurred during this time?.1 Solution. C top. Ψ ψ left. Note C left 1 b. 1 c. 0 3 Problem 3 3. Wht is the first excited stte wvefunction for the one-dimensionl hrmonic oscilltor use generic normliztion constnt for the current purposes? 3b. Set up the eqution for determining the verge kinetic energy for this stte tht of prt 3. 3c. Wht is the verge vlue of x? 3.1 Solution 3. first excited stte: n 1: From pge 11- in hndbook ψ n x A n H n x H n is determined from Tble 11.1 in Hndbook for n 1. Thus, the totl wvefunction with generic normliztion constnt is ψ n1 x A n x
3 Note tht in this cse, the complex conjugte of the wvefunction is the sme s the wvefunction since we re deling with rel functions this is specil cse, be creful for the generl problems you my encounter in the future ψ n1x A n x 3b. verge kinetic energy: KE ψ x KE ˆ ψx ψ x ψx The kinetic energy opertor is KE ˆ h d m Thus, using the bove ressions for the wvefunction nd its complex conjugte: KE ψ x h m d ψ x ψx ψx [ A x ] [ n x h m [ A x n d A n x ] [ A n x x ] ] 3c. Averge vlue of x. x ψ x x ψx ψ x ψx We will consider the numertor nd denomintor individully. The numertor is: ψ x x ψx 4 A n x 3 e x / 4 A n x 3 e x / 3
4 Consider the symmetry of the integrnd, x 3 e x /. This is of odd symmetry. Thus, we cn esily write down the nswer to the integrl s 0! Now, let s just mke sure tht the denomintor is not zero. ψ x ψx 4 A n 8 A n 0 x e x / x e x / The solution to the lst integrl is found in Tble.3 fifth integrl on the left side. Thus, the denomintor is not zero, nd the verge vlue of x is 0 s we would ect intuitively. 4 Extr Credit: 10 points For the n sttionry stte of 1-D prticle in box, wht re the possible vlues of the momentum? Wht is the probbility of ech? Consider the following helpful Euler reltions: 4.1 Solution e ikx e ikx coskx + i sinkx coskx i sinkx For the n wvefunction for 1-D prticle in box, the wvefunction is: ψx π x sin We re sked for the vlues of momentum tht would be observble for this prticle. If the quntum mechnicl opertor for momentum were to operte on this wvefunction, we would see tht the form of the wvefunction given bove is not n eigenfunction of the momentum opertor. Thus, we hve no wy of telling the vlues of momentum we would observe with the wvefunction s given bove. We need to modify it; specificlly, we need to rewrite it s n nsion in the bsis eigenfunctions of the momentum opertor. 4
5 To do this, we cn use the Euler reltions given. Specificlly, we cn subtrct the two functions to obtin: e ikx e ikx coskx i sinkx coskx i sinkx i sinkx Thus, form for the sinkx function in terms of the onentils is: sinkx 1 i e ikx + 1 i eikx The onentils re eigenfunctions of the momentum opertor. Returning to the 1-D prticle in box wvefunction for n, we cn by nlogy: π x sin 1 i e i π x + 1 i e i π x Operting on the individul onentils with the momentum opertor, we obtin for the eigenvlues: π h nd π h Since the mgnitude of the coefficients for the two eigenfunctions in the nsion re equl, their squres re equl. Since the probbility of observing either wvefunction s observble the eigenvlue is equl to tht of the other, the probbility for ech vlue of momentum is
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