Download Practice Problem Set 3 - Physical Chemistry: Quantum Mechanics | CHEM 113A and more Assignments Chemistry in PDF only on Docsity! Chem 113 Fall 07 PROBLEM SET 3 Due Friday 10/26 1. Prove that the time rate of change of the mean value of any observable A is given by ˆ ˆˆ , d A i H A dt ⎡ ⎤= ⎣ ⎦ Assume operators to be independent of time. Hint: once you get started, think time-dependent Schrödinger equation 2. Using the formula derived in question 1, show that A. ˆ ˆ xd x p dt m = B. ˆ xd p V dt x ∂ = − ∂ Interpret these fundamental results in the context of what you know from classical mechanics. NOTE: This is another question taken verbatim from a former midterm. 3. A. Consider a very simple observable, “the number 2”. This is the observable that, in my daily life, whenever I observe it the result is the number 2. The question is what will be the result of such an observation in quantum mechanics. The operator , meaning that the operator is multiplication by the number 2. Show that in quantum mechanics for any normalized state of the system, the average value (sometimes called the expected value) of the operator is the number 2, so it makes good classical correspondence. 2̂ 2= • 2̂ B. It looks like that for any normalized state of the system the operator has the sharp value 2 without any uncertainty. Can you suggest why there is no dispersion in the observed values? 2̂ 4. The operator  , representing the observable A, has two normalized eigenstates 1ψ and 2ψ , belonging to the eigenvalues and , respectively. The operator 1a 2a B̂ , representing the observable B, also has two normalized eigenstates 1φ and 2φ , belonging to the eigenvalues and , respectively. The eigenstates of the two operators are related by the following equations: 1b 2b 1 1 2 1 1 (3 4 ) 5 1 (3 4 ) 5 2 2 ψ φ φ ψ φ φ = + = − A. Observable A is measured, and the value is obtained. What is the state of the system (immediately) after this measurement? 1a B. If B is now measured, what are the possible results, and what are their probabilities? C. Right after the measurement of B, A is measured again. What is the probability of getting ? 1a 5. Suppose we prepare a system in the ground state of the 1D particle-in-a-box Hamiltonian, where the length of the box is L. Imagine that at time zero, the length of the box suddenly doubles. Use the eigenfunction expansion postulate to find the probability that the system (particle) will be found in the ground state of the new box (i.e. a box of length 2L) Hamiltonian. Recall: the expansion postulate states that any arbitrary function (say in 1D – but this is true for any D) can be written as an infinite sum of eigenfunctions of any hermitian operator. Symbolically, this means that if the set of functions nψ satisfies ˆ n n nA aψ ψ= , then any ( )f x can be written as ( ) ( )n n n f x cψ=∑ x
