2. The quantum states of a particle moving freely in a circle of radius r are described by Pn(0) = Cetwno where C is a constant, 0 denotes angle, n = 0, ±1,±2,... is an integer identifying the quantum state of the particle, and wn is constant for a given n. a) Show that Vn(0) satisfies d02 b) Find wn such that Un (0 + 27) = vn(0) c) Find the value of C such that |bn(0)|²d® = 1 d) Show that any two n(0) and vm (0) with m+n satisfy | V(0)vm(0)d® = 0

Q: 10 of 17 If y(t)+(t-r)y(r)dr=t, then y(t)= Su-nyndr3Dt,

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Q: 30) Let n 2 3 and k e Zn. Prove that in Dn. p u = up"-k, %3D

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Q: (2) O show that you Y2,43, and Yu are solutians to NOTE : Geueral Solufion

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Q: 2. Find +he source (メ) Q (x) that the So lutiom U (x it) of k + Qx) 2 - 0 l• Con e

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Q: 7. Find T, N, ar, an at t=n. r(1) = 4cos(1)i–4sin(r)j-3rk

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Q: 4: a) Let X.X., X -N (Hi,6f). Let Y,, Y2 Ym-N (u2,63). .... d (1-a) % CIfor oi when ai. a known.

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