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Unformatted text preview: Physics 3220 – Quantum Mechanics 1 – Fall 2008 Problem Set #4 Due Wednesday, September 17 at 2pm Problem 4.1 : Stationary state in the infinite square well. (20 points) The infinite square well has the potential V ( x ) = 0 , ≤ x ≤ a, (1) = ∞ otherwise , (2) and the (normalized) stationary states were found to be ψ n ( x ) = s 2 a sin nπx a , (3) with energies E n = n 2 π 2 ¯ h 2 2 ma 2 . (4) a) If a wavefunction at time t = 0 is Ψ( x, 0) = ψ n ( x ), write down Ψ( x,t ) at all times. b) Calculate the expectation values h x i , h x 2 i , h p i and h p 2 i for the n th stationary state. Briefly describe the physical meaning of the h x i and h p i results. c) Calculate the standard deviations σ x and σ p , called “uncertainties” in quantum mechanics. One grows much more rapidly than the other as n increases; can you make physical sense of why they behave differently? Think about the values that x and p may take. d) Heisenberg’s Uncertainty Principle states that for any physical wavefunction Ψ, the un- certainties σ x and σ p will always obey σ x σ p ≥ ¯ h 2 . (5) Check Heisenberg’s Uncertainty Principle in the case of the stationary states. Which sta- tionary state is closest to the lower bound of the inequality? Now that we know about how to check expectation values for momentum, it’s time to get some practice, while getting acquainted with the infinite square well and its stationary states. The Uncertainty Principle will appear more later. 1 Problem 4.2 : Non-stationary state in the infinite square well. (20 points) Using the same conventions for the infinite square well as the previous problem, a wavefunc- tion at time t = 0 is Ψ( x, 0) = A ( ψ 2 ( x ) + ψ 3 ( x )) . (6) a) Normalize Ψ( x, 0). There is an easy way and a less easy way: the easy way is to exploit the orthonormality of the ψ n ( x ). b) Determine Ψ( x,t ) and | Ψ( x,t ) | 2 , and write the latter in an explicitly real form (no i ’s anywhere). What is the angular frequency ω of oscillation of | Ψ( x,t ) | 2 , and how is it related to the stationary state energies?to the stationary state energies?...
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This note was uploaded on 02/27/2012 for the course PHYSICS 3220 taught by Professor Stevepollock during the Fall '08 term at Colorado.
- Fall '08