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

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is completely described by a state function Ø(q, t) or ket jØi, which depends on spatial coordinates q and the time t. This function is sometimes also called a state vector or a wave function. The coordinate vector q has components q1, q2, . . . , so that the state function may also be written as Ø(q1, q2, . . . , t). For a particle or system that moves in only one dimension (say along the x-axis), the vector q has only one component and the state vector Ø is a function of x and t: Ø(x, t). For a particle or system in three dimensions, the components of q are x, y, z and Ø is a function of the position vector r and t: Ø(r, t). The state function is single-valued, a continuous function of each of its variables, and square or quadratically integrable. For a one-dimensional system, the quantity Ø\ufffd(x, t)Ø(x, t) is the probabil- ity density for finding the system at position x at time t. In three dimensions, the quantity Ø\ufffd(r, t)Ø(r, t) is the probability density for finding the system at point r at time t. For a multi-variable system, the product Ø\ufffd(q1, q2, . . . , t)Ø(q1, q2, . . . , t) is the probability density that the system has coordi- nates q1, q2, . . . at time t. We show below that this interpretation of Ø \ufffdØ follows from postulate 3. We usually assume that the state function is normal- ized Ø\ufffd(q1, q2, . . . , t)Ø(q1, q2, . . . , t)w(q1, q2, . . .) dq1 dq2 . . . 1 or in Dirac notation hØjØi 1 where the limits of integration are over all allowed values of q1, q2, . . . Physical quantities or observables The second postulate states that a physical quantity or observable is represented in quantum mechanics by a hermitian operator. To every classically defined function A(r, p) of position and momentum there corresponds a quantum- mechanical linear hermitian operator A^(r, ("=i)=). Thus, to obtain the quan- tum-mechanical operator, the momentum p in the classical function is replaced by the operator p^ p^ " i = (3:44) or, in terms of components p^x " i @ @x , p^y " i @ @ y , p^z " i @ @z 86 General principles of quantum theory For multi-particle systems with cartesian coordinates r1, r2, . . . , the classical function A(r1, r2, . . . , p1, p2, . . .) possesses the corresponding operator A^(r1, r2, . . . , ("=i)=1, ("=i)=2, . . .) where =k is the gradient with respect to rk . For non-cartesian coordinates, the construction of the quantum-mechanical operator A^ is more complex and is not presented here. The classical function A is an observable, meaning that it is a physically measurable property of the system. For example, for a one-particle system the Hamiltonian operator H^ corresponding to the classical Hamiltonian function H(r, p) p 2 2m V (r) where p2 p : p p2x p2y p2z , is H^ ÿ " 2 2m =2 V (r) The linear operator H^ is easily shown to be hermitian. Measurement of observable properties The third postulate relates to the measurement of observable properties. Every individual measurement of a physical observable A yields an eigenvalue ºi of the operator A^. The eigenvalues are given by A^jii ºijii (3:45) where jii are the orthonormal eigenkets of A^. Since A^ is hermitian, the eigenvalues are all real. It is essential for the theory that A^ is hermitian because any measured quantity must, of course, be a real number. If the spectrum of A^ is discrete, then the eigenvalues ºi are discrete and the measurements of A are quantized. If, on the other hand, the eigenfunctions jii form a continuous, infinite set, then the eigenvalues ºi are continuous and the measured values of A are not quantized. The set of eigenkets jii of the dynamical operator A^ are assumed to be complete. In some cases it is possible to show explicitly that jii forms a complete set, but in other cases we must assume that property. The expectation value or mean value hAi of the physical observable A at time t for a system in a normalized state Ø is given by hAi hØjA^jØi (3:46) If Ø is not normalized, then the appropriate expression is hAi hØjA^jØihØjØi Some examples of expectation values are as follows 3.7 Postulates of quantum mechanics 87 hxi hØjxjØi hpxi Ø \ufffd\ufffd\ufffd\ufffd "i @@x \ufffd\ufffd\ufffd\ufffdØ * + hri hØjrjØi hpi Ø \ufffd\ufffd\ufffd\ufffd "i = \ufffd\ufffd\ufffd\ufffdØ * + E hHi Ø \ufffd\ufffd\ufffd\ufffdÿ "22m =2 V (r) \ufffd\ufffd\ufffd\ufffdØ * + The expectation value hAi is not the result of a single measurement of the property A, but rather the average of a large number (in the limit, an infinite number) of measurements of A on systems, each of which is in the same state Ø. Each individual measurement yields one of the eigenvalues ºi, and hAi is then the average of the observed array of eigenvalues. For example, if the eigenvalue º1 is observed four times, the eigenvalue º2 three times, the eigenvalue º3 once, and no other eigenvalues are observed, then the expectation value hAi is given by hAi 4º1 3º2 º3 8 In practice, many more than eight observations would be required to obtain a reliable value for hAi. In general, the expectation value hAi of the observable A may be written for a discrete set of eigenfunctions as hAi X i Piºi (3:47) where Pi is the probability of obtaining the value ºi. If the state function Ø for a system happens to coincide with one of the eigenstates jii, then only the eigenvalue ºi would be observed each time a measurement of A is made and therefore the expectation value hAi would equal ºi hAi hijA^jii hijºijii ºi It is important not to confuse the expectation value hAi with the time average of A for a single system. For an arbitrary state Ø at a fixed time t, the ket jØi may be expanded in terms of the complete set of eigenkets of A^. In order to make the following discussion clearer, we now introduce a slightly more complicated notation. Each eigenvalue ºi will now be distinct, so that ºi 6 º j for i 6 j. We let gi be 88 General principles of quantum theory the degeneracy of the eigenvalue ºi and let jiÆi, Æ 1, 2, . . . , gi, be the orthonormal eigenkets of A^. We assume that the subset of kets corresponding to each eigenvalue ºi has been made orthogonal by the Schmidt procedure outlined in Section 3.3. If the eigenkets jiÆi constitute a discrete set, we may expand the state vector jØi as jØi X i Xgi Æ1 ciÆjiÆi (3:48) where the expansion coefficients ciÆ are ciÆ hiÆjØi (3:49) The expansion of the bra vector hØj is, therefore, given by hØj X j Xg j â1 c\ufffdjâh jâj (3:50) where the dummy indices i and Æ have been replaced by j and â. The expectation value of A^ is obtained by substituting equations (3.48) and (3.50) into (3.46) hAi X j Xg j â1 X i Xgi Æ1 c\ufffdjâciÆh jâjA^jiÆi X j Xg j â1 X i Xgi Æ1 c\ufffdjâciÆºih jâjiÆi X i Xgi Æ1 jciÆj2ºi (3:51) where we have noted that the kets jiÆi are orthonormal, so that h jâjiÆi äijäÆâ A comparison of equations (3.47) and (3.51) relates the probability Pi to the expansion coefficients ciÆ Pi Xgi Æ1 jciÆj2 Xgi Æ1 jhiÆjØij2 (3:52) where equation (3.49) has also been introduced. For the case where ºi is non- degenerate, the index Æ is not needed and equation (3.52) reduces to Pi jcij2 jhijØij2 For a continuous spectrum of eigenkets with non-degenerate eigenvalues, it is more convenient to write the eigenvalue equation (3.45) in the form A^jºi ºjºi where º is now a continuous variable and jºi is the eigenfunction whose eigenvalue is º. The expansion of the state vector Ø becomes 3.7 Postulates of quantum mechanics 89 jØi c(º)jºi dº where c(º) hºjØi and the expectation value of A takes the form hAi jc(º)j2 º dº (3:53) If dPº is the probability of obtaining a value of A between º and º dº, then equation