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Test3-FS01 - Phys 208 Theoretical Physics Test 3 Nov 19...

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Phys 208 – Theoretical Physics – Test 3 -- Nov. 19, 2001 1.(25 pts) Given . F x y z xy yz z x ( , , ) sin = + + a) Find at (0, 1, 2). r F b) Find a unit vector in the direction of most rapid increase of F at (0, 1, 2). c) Find the directional derivative of F at (0, 1, 2) in the direction . 2 2 $ $ $ i j k + 2.(25 pts) One of the hydrogen radial wave functions is given by , where and is the Bohr radius. R r N re r a 21 21 2 0 ( ) / = 0 < < ∞ r a 0 a) Determine the constant by normalizing this radial wave function. N 21 b) Determine the most probable value of r for this radial wave function. 3.(25 pts) Calculate the divergence of , r A r z r r r zz ( , , ) ( sin ) $ sin cos $ $ θ θ θ θ θ = + + + 2 3 2 which is given in cylindrical coordinates . ( , , ) r z θ 4.(25 pts) The normalized ground and first excited state wave functions of the 1-dimensional harmonic oscillator are given by: and , ( ) ψ π 0 1 4 2 2 ( ) / / x e a ax = ( ) ψ π 1 1 4 2 2 2 ( ) / / x a xe a ax = where a is a constant and x is defined to be between !4 and + 4 . These states have energies and . Consider the time-dependent wave function defined as: E 0 2 = h ω / E 1 3 2 = h ω / . [ ] Ψ ( , ) ( ) ( ) / / x t x e x e iE t iE t = +
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