Question

Feynman Problem: A very important quantum model that is used ubiquitously in physics and in chemistry is a quanton on a ring, a modification of quanton in a box i.e. the box wraps in on itself. The time-independent Schrödinger equation used to describe quanton on a ring is: h2 a2 mazy(8) + V(s)(s) = Ev(s) where R is the radius of the ring, o is the angle along the ring the quanton's wavefunction is being evaluated and s = Ro, the arc length transversed along the ring. a) Argue that alas can be written as R-la/ao if the quanton stays on a constant radius R. Hence we may write y(s) as (o). b) Assume the the quanton interacts with its environment with a V(S) = 0i.e. it is free to move anywhere along the ring. Show that y ) = N cos(b) is a solution to the differential equation. N is a normalization constant and b is a constant whose value we will determine next. c) Calculate the normalization constant N and b by insisting your wavefunction for the quanton on the ring be single valued i.e. that (0) = v(0 + 2n) and that the integral of (0)| around the ring be unity i.e. 1. d) Calculate the allowed energy values of the system and comment on whether these values are quantized for the ring. How do these energy values compare to quanton in a box?

Answer 1

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=> dq (0) 4x2 4.09 ) 20(Ezmest) it solulim as given as Q6 ) = N(os (26) sonce cos (+B) = COS A GSB - SMA SMB 5 N Cos (+) = N Coc (b$+622) - neos (30) = NGOs (60) as (621) - also to sombra B - -) N cos 6+) = Nas (bu) Bas (201) → as (296) = of (son 296=0) bi catherine in (= 9.40] Normalization 9 lip Jet 42 j s vrtca 424) déb = 1 Ja ma ei auta nge)df-41 *{* = 4( 424) Ne con = 1 =) W7 = 14 ) N=1 / 21 0 Scanned with CamScanner

د E ایا 1. Lue Ime2 Quantised Energy ) En = 42 . (his inlegs) Where as for quantom in a book we have Enanth Scanned with CamScanner