HW10 - Physics 3220 Quantum Mechanics 1 Fall 2008 Problem...

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Physics 3220 – Quantum Mechanics 1 – Fall 2008 Problem Set #10 Due Wednesday, Nov 5 at 2pm Problem 10. 1 Generalized Uncertainty [15 pts] A) If operators  and B are Hermitian, what must be true about the constant α in order to ensure that α A, B is also a Hermitian operator? This result is relevant to the generalized uncertainty principle (GUP, Griffiths Eq 3.62): setting α =1/2i, show that the right-hand-side of the GUP will indeed always be real and nonnegative. (If the right side weren't real and nonnegative, would the GUP be of any use at all?) B) Show that the generalized uncertainty principle for x and H (position and Hamiltonian) is σ x s H h 2m p . (1) What can you deduce about <p> for any stationary state? Is this uncertainty principle useful for stationary states? Lastly, show that result (1) also follows immediately from Griffiths' form of the "energy-time uncertainty principle", namely his Eq 3.72 (with an obvious choice for Q in that equation) Problem 10. 2 Time dependence [15 pts] A) Apply Griffiths Eq 3.71 (page 115) to the following special cases: Q=1, Q=H, and Q=p. (Note that you already DID this for Q=x on last week's homework.) In each case (including the Q=x case from last week), comment on the result, with particular reference to Griffiths Equations 1.27, 1.33, 1.38, and conservation of energy (which Griffiths talks about on the bottom of page 37) B) Consider a particle which is a 50-50 mixture of ground state and first excited states in the infinite square well, Ψ ( x ,0) = A ( u 1 ( x ) + u 2 ( x )) . (Choose A to normalize the state. What is the time dependence of this state?) Compute <H> and <H 2 >, and thus the uncertainty in energy, H . Simplify your result as much as possible - in particular, write your final expression so it is easy to see how big it is compared to (E 2 -E 1 ) Problem 10. 3 Uncertainty principle for small but still macro objects. [15 pts] We never notice the Uncertainty Principle ( x p h ) for macroscopic objects, because Planck's constant is so small. Let's see how big an effect the Uncertainty Principle produces for an object that is very small, but still large compared to atoms. Consider a 1-micrometer diameter droplet of oil suspended in a vacuum chamber. (You can suspend an oil droplet by charging it and applying an electric field, like Millikan did in his famous experiment)
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HW10 - Physics 3220 Quantum Mechanics 1 Fall 2008 Problem...

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