Tag: Griffith’s Introduction to Quantum Mechanics 3rd Edition

Problem 4.1 (a) Work out all of the canonical commutation relations for components of the operators r and p: , , , , and so on.Answer: where the indices stand for , , or , and , , and . (b) Confirm the three-dimensional version of Ehrenfest’s theorem, (Each of these, of course, stands for…

Problem 2.8 A particle of mass m in the infinite square well (of width starts outin the state for some constant , so it is (at ) equally likely to be found at any point in the left half of the well. What is the probability that a measurement of the energy (at some later…

Problem 2.7 A particle in the infinite square well has the initial wave function (a) Sketch , and determine the constant (b) Find .(c) What is the probability that a measurement of the energy would yield the value ?(d) Find the expectation value of the energy, using Equation 2.2. Solution: Peoblem 2.7 Solution (Download)

Problem 2.6 Although the overall phase constant of the wave function is of no physical significance (it cancels out whenever you calculate a measurable quantity), the relative phase of the coefficients in Equation 2.17 does matter. For example, suppose we change the relative phase of and in Problem 2.5: where is some constant. Find ,…

Problem 2.5 A particle in the infinite square well has as its initial wave function an even mixture of the first two stationary states: (a) Normalize . (That is, find This is very easy, if you exploit the orthonormality of and . Recall that, having normalized at , you can rest assured that it stays…

Problem 2.4 Calculate , , , , , and , for the nth stationary state of the infinite square well. Check that the uncertainty principle is satisfied. Which state comes closest to the uncertainty limit? Solution: Problem 2.4 Solution (Download)

Problem 2.3 Show that there is no acceptable solution to the (time-independent) Schrödinger equation for the infinite square well with or . (This is a special case of the general theorem in Problem 2.2, but this time do it by explicitly solving the Schrödinger equation, and showing that you cannot satisfy the boundary conditions.) Solution:…

Problem 2.2 Show that must exceed the minimum value of , for every normalizable solution to the time-independent Schrödinger equation. What is the classical analog to this statement? Hint: Rewrite Equation 2.5 in the form if , then and its second derivative always have the same sign—argue that such a function cannot be normalized. Solution:…

Problem 2.1 Prove the following three theorems: (a) For normalizable solutions, the separation constant must be real. Hint: Write (in Equation 2.7) as (with and real), and show that if Equation 1.20 is to hold for all , must be zero. (b) The time-independent wave function can always be taken to be real (unlike ,…

Problem 2.9 For the wave function in Example 2.2, find the expectation value ofH, at time t=0, the “old fashioned” way: Compare the result we got in Example 2.3. Note: Because is independent of time, there is no loss of generality in using t=0. Solution: Problem 2.9 Solution (Download)