[理学]量子化学课程习题及标准答案.doc

量子化学习题及标准答案 Chapter 01 1. A certain one-particle, one-dimensional system has , where a and b are constants and m is the particle’s mass. Find the potential-energy function V for this system. (Hint: Use the time-dependent Schrodinger equation.) Solution：As ?(x,t) is known, we can derive the corresponding derivatives. According to time-dependent Schroedinger equation, substituting into the derivatives, we get 2. At a certain instant of time, a one-particle, one-dimensional system has , where b = 3.000 nm. If a measurement of x is made at this time in the system, find the probability that the result (a) lies between 0.9000 nm and 0.9001 nm (treat this interval as infinitesimal); (b) lies between 0 and 2 nm (use the table of integrals, if necessary). (c) For what value of x is the probability density a minimum? (There is no need to use calculus to answer this.) (d) Verify that is normalized. Solution：a) The probability of finding an particle in a space between x and x+dx is given by b) c) Clearly, the minimum of probability density is at x=0, where the probability density vanishes. d) 3. A one-particle, one-dimensional system has the state function where a is a constant and c = 2.000 ?. If the particle’s position is measured at t = 0, estimate the probability that the result will lie between 2.000 ? and 2.001 ?. Solution：when t=0, the wavefunction is simplified as Chapter 02 1. Consider an electron in a one-dimensional box of length 2.000? with the left end of the box at x = 0. (a) Suppose we have one million of these systems, each in the n = 1 state, and we measure the x coordinate of the electron in each system. About how many times will the electron be found between 0.600 ? and 0.601 ?? Consider the interval to be infinitesimal. Hint: Check whether your calculator is set to degrees or radians. (b) Suppose we have a large number of these systems, each in the n =1 state, and we measure the x coordinate of the electron in each system and find the ele