Download Quantum Mechanics Problem Set: Infinite Square Well and Uncertainty Principle and more Assignments Quantum Mechanics in PDF only on Docsity! 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 , 0 ≤ x ≤ a , (1) = ∞ otherwise , (2) and the (normalized) stationary states were found to be ψn(x) = √ 2 a sin ( nπx a ) , (3) with energies En = n2π2h̄2 2ma2 . (4) a) If a wavefunction at time t = 0 is Ψ(x, 0) = ψn(x), write down Ψ(x, t) at all times. b) Calculate the expectation values 〈x〉, 〈x2〉, 〈p〉 and 〈p2〉 for the nth stationary state. Briefly describe the physical meaning of the 〈x〉 and 〈p〉 results. c) Calculate the standard deviations σx and σp, 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 σx and σp will always obey σxσp ≥ h̄ 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 (ψ2(x) + ψ3(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 ψn(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? c) Calculate 〈x〉 and 〈p〉 as functions of time and express them in terms of ω. Hint: once you have 〈x〉 as a function of time, there is a shortcut to calculating 〈p〉. d) If you measured the energy of this particle, what are the values you might get, and what is the probability of each of them? Find the expectation values 〈H〉 and 〈H2〉, which are the same thing as 〈E〉 and 〈E2〉, and the uncertainty in the energy σE. Not all states are stationary states, even though all states can be built from stationary states, and in fact non-stationary states are much more interesting — where by “interesting” we mean “something actually happens”. Problem 4.3: Expanding infinite square well and time evolution. (30 points) For times t < 0, a particle is sitting in the ground state of an infinite square well of length 1/2, 0 ≤ x ≤ 1/2. At t = 0, the experimenter causes the well to be doubled in size to occupy 0 ≤ x ≤ 1. This problem requires some numerical work, with software such as Mathematica or Matlab. Posted on the web site are two sample Mathematica notebooks you may find useful if you don’t know much about it, but you are free to use other software if you wish. a) Assume that the change in the size of the well does not affect the wavefunction at the instant t = 0; write down the wavefunction Ψ(x, t = 0). (Think carefully about what is happening in different ranges of x.) b) The wavefunction can be written as a linear combination of stationary states of the larger well, Ψ(x, t = 0) = ∑∞ n=1 cnψn(x), where ψn are the stationary states of the well of size 1. Determine a formula for the cn. To check that this infinite series is approximating Ψ(x, t = 0), plot the terms up to n = 5 and compare the result to the exact graph. (Going to higher n should make the comparison better, but n = 5 will be good enough for us.) 2
