量子化学课程习题及标准答案

量子化学习及答案 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 ψ(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 electron between 0.700 ? and 0.701 ? in 126 of the measurements. In about how many measurements will the electron be found between 1.000 ? and 1.001 ?? Solution: a) In a 1D box, the energy and wave-function of a micro-system are given by therefore, the probability density of finding the electron between 0.600 and 0.601 ? is b) From the definition of probability, the probability of finding an electron between x and x+dx is given by As the number of measurements of finding the electron between 0.700 and 0.701 ? is known, the number of system is 2. When a particle of mass 9.1*10-28 g in a certain one-dimensional box goes from the n = 5 level to the n = 2 level, it emits a photon of frequency 6.0*1014 s-1. Find the length of the box. Solution. 3. An electron in a stationary state of a one-dimensional box of length 0.300 nm emits a photon of frequency 5.05*1015 s-1. Find the initial and final quantum numbers for this transition. Solution: 4. For the particle in a one-dimensional box of length l, we could have put the coordinate origin at the center of the box. Find the wave functions and energy levels for this choice of origin. Solution: The wavefunction for a particle in a one-dimernsional box can be written as If the coordinate origin is defined at the center of the box, the boundary conditions are given as Combining Eq1 with Eq2, we get Eq3 leads to A=0, or =0. We will discuss both situations in the following section. If A=0, B must be non-zero number otherwise the wavefunction vanishes. If A≠0 5. For an electron in a certain rectangular well with a depth of 20.0 eV, the lowest energy lies 3.00 eV above the bottom of the well. Find the width of this well. Hint: Use tanθ = sinθ/cosθ Solution: For the particle in a certain rectangular well, the E fulfill with Substituting into the V and E, we get Chapter 03 1. If f (x) = 3x2 f(x) + 2xd f /dx, give an expression for . Solution： Extracting f(x) from the known equation leads to the expression of A 2. (a) Show that ( + )2 = ( + )2 for any two operators. (b) Under what conditions is ( + )2 equal to 2+2 + 2? Solution: a) b) If and only if A and B commute, ( + )2 equals to 2+2 + 2 3. If = d2/dx2 and = x2, find (a) x3; (b) x3; (c) f(x); (d) f(x) Solution: a) b) c) d) 4. Classify these operators as linear or nonlinear: (a) 3x2d2/dx2; (b) ( )2; (c) ∫ dx; (d) exp; (e) . Solution: Linear operator is subject to the following condition. a)  Linear b)  Nonlinear c)  Linear d)  Nonlinear e)  Linear 5. The Laplace transform operator is defined by (a) Is linear? (b) Evaluate (1). (c) Evaluate eax, assuming that p>a. Solution: a) L is a linear operator b) c) 6. We define the translation operator by f (x) = f (x + h). (a) Is a linear operator? (b) Evaluate ( )x2. Solution: a) The translation operator is linear operator