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

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(3.47) is replaced by hAi º dPº and we see that dPº jc(º)j2 dº jhºjØij2 dº The probability dPº is often written in the form dPº r(º) dº where r(º) is the probability density of obtaining the result º and is given by r(º) jc(º)j2 jhºjØij2 In terms of the probability density, equation (3.53) becomes hAi ºr(º) dº (3:54) In some applications to physical systems, the eigenkets of A^ possess a partially discrete and a partially continuous spectrum, in which case equations (3.51) and (3.53) must be combined. The scalar product hØjØi may be evaluated from equations (3.48) and (3.50) as hØjØi X j Xg j â1 X i Xgi Æ1 c\ufffdjâciÆh jâjiÆi X i Xgi Æ1 jciÆj2 X i Xgi Æ1 jhiÆjØij2 X i Pi Since the state vector Ø is normalized, this expression givesX i Pi 1 Thus, the sum of the probabilities Pi equals unity as it must from the definition of probability. For a continuous set of eigenkets, this relationship is replaced by 90 General principles of quantum theory dPº r(º) dº 1 As an example, we consider a particle in a one-dimensional box as discussed in Section 2.5. Suppose that the state function Ø(x) for this particle is time- independent and is given by Ø(x) C sin5 ðx a \ufffd \ufffd , 0 < x < a where C is a constant which normalizes Ø(x). The eigenfunctions jni and eigenvalues En of the Hamiltonian operator H^ are jni 2 a r sin nðx a \ufffd \ufffd , En n 2 h2 8ma2 , n 1, 2, . . . Obviously, the state function Ø(x) is not an eigenfunction of H^ . Following the general procedure described above, we expand Ø(x) in terms of the eigenfunc- tions jni. This expansion is the same as an expansion in a Fourier series, as described in Appendix B. As a shortcut we may use equations (A.39) and (A.40) to obtain the identity sin5\u141 1 16 (10 sin \u141ÿ 5 sin 3\u141 sin 5\u141) so that the expansion of Ø(x) is Ø(x) C 16 10 sin ðx a \ufffd \ufffd ÿ 5 sin 3ðx a \ufffd \ufffd sin 5ðx a \ufffd \ufffd" # C 16 a 2 r (10j1i ÿ 5j3i j5i) A measurement of the energy of a particle in state Ø(x) yields one of three values and no other value. The values and their probabilities are E1 h 2 8ma2 , P1 10 2 102 52 12 100 126 0:794 E3 9h 2 8ma2 , P3 5 2 126 0:198 E5 25h 2 8ma2 , P5 1 2 126 0:008 The sum of the probabilities is unity, P1 P3 P5 0:794 0:198 0:008 1 The interpretation that the quantity Ø\ufffd(q1, q2, . . . , t)Ø(q1, q2, . . . , t) is the probability density that the coordinates of the system at time t are 3.7 Postulates of quantum mechanics 91 q1, q2, . . . may be shown by comparing equations (3.46) and (3.54) for A equal to the coordinate vector q hqi hØjqjØi Ø\ufffd(q)Ø(q)q dô hqi r(q)q dô For these two expressions to be mutually consistent, we must have r(q) Ø\ufffd(q)Ø(q) Thus, this interpretation of Ø\ufffdØ follows from postulate 3 and for this reason is not included in the statement of postulate 1. Collapse of the state function The measurement of a physical observable A gives one of the eigenvalues ºn of the operator A^. As stated by the fourth postulate, a consequence of this measurement is the sudden change in the state function of the system from its original form Ø to an eigenfunction or linear combination of eigenfunctions of A^ corresponding to ºn. At a fixed time t just before the measurement takes place, the ket jØi may be expanded in terms of the eigenkets jiÆi of A^, as shown in equation (3.48). If the measurement gives a non-degenerate eigenvalue ºn, then immediately after the measurement the system is in state jni. The state function Ø is said to collapse to the function jni. A second measurement of A on this same system, if taken immediately after the first, always yields the same result ºn. If the eigenvalue ºn is degenerate, then right after the measurement the state function is some linear combination of the eigenkets jnÆi, Æ 1, 2, . . . , gn. A second, immediate measurement of A still yields ºn as the result. From postulates 4 and 5, we see that the state function Ø can change with time for two different reasons. A discontinuous change in Ø occurs when some property of the system is measured. The state of the system changes suddenly from Ø to an eigenfunction or linear combination of eigenfunctions associated with the observed eigenvalue. An isolated system, on the other hand, undergoes a continuous change with time in accordance with the time-dependent Schro¨- dinger equation. Time evolution of the state function The fifth postulate stipulates that the time evolution of the state function Ø is determined by the time-dependent Schro¨dinger equation 92 General principles of quantum theory i" @Ø @ t H^Ø (3:55) where H^ is the Hamiltonian operator of the system and, in general, changes with time. However, in this book we only consider systems for which the Hamiltonian operator is time-independent. To solve the time-dependent Schro¨- dinger equation, we express the state function Ø(q, t) as the product of two functions Ø(q, t) \u142(q)÷(t) (3:56) where \u142(q) depends only on the spatial variables and ÷(t) depends only on the time. In Section 2.4 we discuss the procedure for separating the partial differential equation (3.55) into two differential equations, one involving only the spatial variables and the other only the time. The state function Ø(q, t) is then shown to be Ø(q, t) \u142(q)eÿiEt=" (3:57) where E is the separation constant. Since it follows from equation (3.57) that jØ(q, t)j2 j\u142(q)j2 the probability density is independent of the time t and Ø(q, t) is a stationary state. The spatial differential equation, known as the time-independent Schro¨din- ger equation, is H^\u142(q) E\u142(q) Thus, the spatial function \u142(q) is actually a set of eigenfunctions \u142n(q) of the Hamiltonian operator H^ with eigenvalues En. The time-independent Schro¨din- ger equation takes the form H^\u142n(q) En\u142n(q) (3:58) and the general solution of the time-dependent Schro¨dinger equation is Ø(q, t) X n cn\u142n(q)e ÿiEn t=" (3:59) where cn are arbitrary complex constants. The appearance of the Hamiltonian operator in equation (3.55) as stipulated by postulate 5 gives that operator a special status in quantum mechanics. Knowledge of the eigenfunctions and eigenvalues of the Hamiltonian operator for a given system is sufficient to determine the stationary states of the system and the expectation values of any other dynamical variables. We next address the question as to whether equation (3.59) is actually the most general solution of the time-dependent Schro¨dinger equation. Are there other solutions that are not expressible in the form of equation (3.59)? To 3.7 Postulates of quantum mechanics 93 answer that question, we assume that Ø(q, t) is any arbitrary solution of the parital differential equation (3.55). We suppose further that the set of functions \u142n(q) which satisfy the eigenvalue equation (3.58) is complete. Then we can, in general, expand Ø(q, t) in terms of the complete set \u142n(q) and obtain Ø(q, t) X n an(t)\u142n(q) (3:60) The coefficients an(t) in the expansion are given by an(t) h\u142n(q)jØ(q, t)i (3:61) and are functions of the time t, but not of the coordinates q. We substitute the expansion (3.60) into the differential equation (3.55) to obtainX n " i @an(t) @ t an(t) H^ \ufffd \ufffd \u142n(q) X n " i @an(t) @ t Enan(t) \ufffd \ufffd \u142n(q) 0 (3:62) where we have also noted that the functions \u142n(q) are eigenfunctions of H^ in accordance with equation (3.58). We next multiply equation (3.62) by \u142\ufffdk (q), the complex conjugate of one of the eigenfunctions of the orthogonal set, and