# Principles of Quantum Mechanics as Applied to Chemistry and Chemical Physics

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this derivation does not apply to a particle in a potential field. The position, momentum, and energy are all dynamical quantities and consequently possess quantum-mechanical operators from which expectation values at any given time may be determined. Time, on the other hand, has a unique role in non-relativistic quantum theory as an independent variable; dynamical quantities are functions of time. Thus, the \u2018uncertainty\u2019 in time cannot be related to a range of expectation values. To obtain the energy-time uncertainty principle for a particle in a time- independent potential field, we set A^ equal to H^ in equation (3.81) (\u2dcE)(\u2dcB) > 1 2 jh[ H^ , B^]ij where \u2dcE is the uncertainty in the energy as defined by (3.75) with A^ H^ . Substitution of equation (3.72) into this expression gives (\u2dcE)(\u2dcB) > " 2 \ufffd\ufffd\ufffd\ufffd dhBidt \ufffd\ufffd\ufffd\ufffd (3:86) In a short period of time \u2dct, the change in the expectation value of B is given by \u2dcB dhBi dt \u2dct When this expression is combined with equation (3.86), we obtain the desired result (\u2dcE)(\u2dct) > " 2 (3:87) We see that the energy and time obey an uncertainty relation when \u2dct is defined as the period of time required for the expectation value of B to change by one standard deviation. This definition depends on the choice of the dynamical variable B so that \u2dct is relatively larger or smaller depending on that choice. If dhBi=dt is small so that B changes slowly with time, then the period \u2dct will be long and the uncertainty in the energy will be small. 3.11 Heisenberg uncertainty principle 103 Conversely, if B changes rapidly with time, then the period \u2dct for B to change by one standard deviation will be short and the uncertainty in the energy of the system will be large. Problems 3.1 Which of the following operators are linear? (a) p (b) sin (c) xD^x (d) D^xx 3.2 Demonstrate the validity of the relationships (3.4a) and (3.4b). 3.3 Show that [A^, [B^, C^]] [B^, [C^, A^]] [C^, [A^, B^]] 0 where A^, B^, and C^ are arbitrary linear operators. 3.4 Show that (D^x x)(D^x ÿ x) D^2x ÿ x2 ÿ 1. 3.5 Show that xeÿx 2 is an eigenfunction of the linear operator (D^2x ÿ 4x2). What is the eigenvalue? 3.6 Show that the operator D^2x is hermitian. Is the operator iD^ 2 x hermitian? 3.7 Show that if the linear operators A^ and B^ do not commute, the operators (A^B^ B^A^) and i[A^, B^] are hermitian. 3.8 If the real normalized functions f (r) and g(r) are not orthogonal, show that their sum f (r) g(r) and their difference f (r)ÿ g(r) are orthogonal. 3.9 Consider the set of functions \u1421 eÿx=2, \u1422 xeÿx=2, \u1423 x2eÿx=2, \u1424 x3eÿx=2, defined over the range 0 < x <1. Use the Schmidt orthogonalization procedure to construct from the set \u142i an orthogonal set of functions with w(x) 1. 3.10 Evaluate the following commutators: (a) [x, p^x] (b) [x, p^ 2 x] (c) [x, H^] (d) [ p^x, H^] 3.11 Evaluate [x, p^3x] and [x 2, p^2x] using equations (3.4). 3.12 Using equation (3.4b), show by iteration that [x n, p^x] i"nx nÿ1 where n is a positive integer greater than zero. 3.13 Show that [ f (x), p^x] i" d f (x) dx 3.14 Calculate the expectation values of x, x2, p^, and p^2 for a particle in a one- dimensional box in state \u142n (see Section 2.5). 3.15 Calculate the expectation value of p^4 for a particle in a one-dimensional box in state \u142n. 3.16 A hermitian operator A^ has only three normalized eigenfunctions \u1421, \u1422, \u1423, with corresponding eigenvalues a1 1, a2 2, a3 3, respectively. For a particular state ö of the system, there is a 50% chance that a measure of A produces a1 and equal chances for either a2 or a3. 104 General principles of quantum theory (a) Calculate hAi. (b) Express the normalized wave function ö of the system in terms of the eigenfunctions of A^. 3.17 The wave function Ø(x) for a particle in a one-dimensional box of length a is Ø(x) C sin7 ðx a \ufffd \ufffd ; 0 < x < a where C is a constant. What are the possible observed values for the energy and their respective probabilities? 3.18 If j\u142i is an eigenfunction of H^ with eigenvalue E, show that for any operator A^ the expectation value of [ H^ , A^] vanishes, i.e., h\u142j[ H^ , A^]j\u142i 0 3.19 Derive both of the Ehrenfest theorems using equation (3.72). 3.20 Show that \u2dcH\u2dcx > " 2m h p^xi Problems 105 4 Harmonic oscillator In this chapter we treat in detail the quantum behavior of the harmonic oscillator. This physical system serves as an excellent example for illustrating the basic principles of quantum mechanics that are presented in Chapter 3. The Schro¨dinger equation for the harmonic oscillator can be solved rigorously and exactly for the energy eigenvalues and eigenstates. The mathematical process for the solution is neither trivial, as is the case for the particle in a box, nor excessively complicated. Moreover, we have the opportunity to introduce the ladder operator technique for solving the eigenvalue problem. The harmonic oscillator is an important system in the study of physical phenomena in both classical and quantum mechanics. Classically, the harmonic oscillator describes the mechanical behavior of a spring and, by analogy, other phenomena such as the oscillations of charge flow in an electric circuit, the vibrations of sound-wave and light-wave generators, and oscillatory chemical reactions. The quantum-mechanical treatment of the harmonic oscillator may be applied to the vibrations of molecular bonds and has many other applica- tions in quantum physics and field theory. 4.1 Classical treatment The harmonic oscillator is an idealized one-dimensional physical system in which a single particle of mass m is attracted to the origin by a force F proportional to the displacement of the particle from the origin F ÿkx (4:1) The proportionality constant k is known as the force constant. The minus sign in equation (4.1) indicates that the force is in the opposite direction to the direction of the displacement. The typical experimental representation of the oscillator consists of a spring with one end stationary and with a mass m 106 attached to the other end. The spring is assumed to obey Hooke\u2019s law, that is to say, equation (4.1). The constant k is then often called the spring constant. In classical mechanics the particle obeys Newton\u2019s second law of motion F ma m d 2x dt2 (4:2) where a is the acceleration of the particle and t is the time. The combination of equations (4.1) and (4.2) gives the differential equation d2x dt2 ÿ k m x for which the solution is x A sin(2ðít b) A sin(øt b) (4:3) where the amplitude A of the vibration and the phase b are the two constants of integration and where the frequency í and the angular frequency ø of vibration are related to k and m by ø 2ðí k m r (4:4) According to equation (4.3), the particle oscillates sinusoidally about the origin with frequency í and maximum displacement \ufffdA. The potential energy V of a particle is related to the force F acting on it by the expression F ÿ dV dx Thus, from equations (4.1) and (4.4), we see that for a harmonic oscillator the potential energy is given by V 1 2 kx2 1 2 mø2x2 (4:5) The total energy E of the particle undergoing harmonic motion is given by E 1 2 mv2 V 1 2 mv2 1 2 mø2x2 (4:6) where v is the instantaneous velocity. If the oscillator is undisturbed by outside forces, the energy E remains fixed at a constant value. When the particle is at maximum displacement from the origin so that x \ufffdA, the velocity v is zero and the potential energy is a maximum. As jxj decreases, the potential decreases and the velocity increases keeping E constant. As the particle crosses the origin