Quantum Mechanics Homework Set 4: Coefficients, Uncertainty, and Energy Eigenvalues, Assignments of Health sciences

Download Quantum Mechanics Homework Set 4: Coefficients, Uncertainty, and Energy Eigenvalues and more Assignments Health sciences in PDF only on Docsity! PHY 389K Quantum Mechanics, Homework Set 4 Solutions Matthias Ihl 02/24/2008 Note: I will post updated versions of the homework solutions on my home- page: http://zippy.ph.utexas.edu/~msihl/teaching.html 1 Problem 1 (a) Let |α〉 = ∑ n cn|n〉 where cn = 〈n|α〉. From the following two facts in different point of view, 〈n|a|α〉 = α〈n|α〉 = αcn and 〈n|a|α〉 = (a†|n〉)∗|α〉 = √ n + 1〈n + 1|α〉 = √ n + 1cn+1, we get a recursion relation cn+1 = α√ n + 1 cn ⇒ cn = αn−1√ n! c0 To get c0 use the normalization condition 〈α|α〉 = 1 and we get |c0|2 ∑ n |α|2n−2 n! = 1 ⇒ |c0|2e|α| 2 /|α|2 = 1 1 ⇒ |c0| = |α|e−|α| 2/2 Set c0 = αe −|α|2/2. Then, |α〉 = e−|α|2/2 ∑ n αn√ n! |n〉 (b) From the operator relation x = √ ~ 2mω (a + a†) p = i √ m~ω 2 (−a + a†) we get 〈x〉 = 〈α|x|α〉 = √ ~ 2mω (α+α∗) 〈p〉 = 〈α|p|α〉 = i √ m~ω 2 (−α+α∗) (1) To compute 〈x2〉 and 〈p2〉 from the operator relation x2 = ~ 2mω (aa+aa†+a†a+a†a†) p2 = −m~ω 2 (aa−aa†−a†a+a†a†) we need to know 〈α|a†a†|α〉 and 〈α|aa†|α〉. •a†a†|α〉 = a†e−|α|2/2 ∑ n αn√ n! √ n + 1|n+1〉 = e−|α|2/2 ∑ n αn√ n! √ n + 1 √ n + 2|n+2〉 ⇒ 〈α|a†a†|α〉 = e−|α|2 ∑ n=2 |α|n−2(α∗)2 (n − 2)! = (α ∗)2 •|aa†|α〉 = e−|α|2/2 ∑ n αn√ n! n + 1|n〉 ⇒ 〈α|aa†|α〉 = |α|2 + 1 Therefore, 〈x2〉 = 〈α|x2|α〉 = ~ 2mω (α2 + 2|α|2 + (α∗)2 + 1) = ~ 2mω ((α + α∗)2 + 1) (2)