(h/2)2d/ or, alternatively between and-i(h/2π)ddx These connections between physical properties(energy E and momentum p)and differential operators are some of the unusual features of quantum mechanics The above discussion about waves and quantized wavelengths as well as the bservations about the wave equation and differential operators are not meant to provide or even suggest a derivation of the Schrodinger equation. Again the scientists who invented quantum mechanics did not derive its working equations. Instead, the equations and rules of quantum mechanics have been postulated and designed to be consistent with laboratory observations. My students often find this to be disconcerting because they are hoping and searching for an uderlying fundamental basis from which the basic laws of quantum mechanics follows logically. I try to remind them that this is not how theory works. Instead, one uses experimental observation to postulate a rule or equation or theory, and one then tests the theory by making predictions that can be tested by further experiments. If the theory fails, it must be "refined", and this process continues until one has a better and better theory. In this sense, quantum mechanics, with all of its unusual mathematical constructs and rules, should be viewed as arising from the imaginations of scientists who tried to invent a theory that was consistent with experimental data and which could be used to predict things that could then be tested in the laboratory. Thus far
16 p 2 and – (h/2p) 2 d 2 /dx2 or, alternatively, between p and – i (h/2p) d/dx. These connections between physical properties (energy E and momentum p) and differential operators are some of the unusual features of quantum mechanics. The above discussion about waves and quantized wavelengths as well as the observations about the wave equation and differential operators are not meant to provide or even suggest a derivation of the Schrödinger equation. Again the scientists who invented quantum mechanics did not derive its working equations. Instead, the equations and rules of quantum mechanics have been postulated and designed to be consistent with laboratory observations. My students often find this to be disconcerting because they are hoping and searching for an uderlying fundamental basis from which the basic laws of quantum mechanics follows logically. I try to remind them that this is not how theory works. Instead, one uses experimental observation to postulate a rule or equation or theory, and one then tests the theory by making predictions that can be tested by further experiments. If the theory fails, it must be “refined”, and this process continues until one has a better and better theory. In this sense, quantum mechanics, with all of its unusual mathematical constructs and rules, should be viewed as arising from the imaginations of scientists who tried to invent a theory that was consistent with experimental data and which could be used to predict things that could then be tested in the laboratory. Thus far
this theory has proven to be reliable, but, of course, we are always searching for a"new and improved" theory that describes how small light particles move If it helps you to be more accepting of quantum theory, I should point out that the quantum description of particles will reduce to the classical Newton description under certain circumstances. In particular, when treating heavy particles(e.g, macroscopic masses and even heavier atoms), it is often possible to use Newton dynamic will discuss in more detail how the quantum and classical dynamics sometimes coincide (in which case one is free to use the simpler Newton dynamics ). So, let us now move on to look at this strange Schrodinger equation that we have been digressing about for so . The Schrodinger equation and Its Components It has been well established that electrons moving in atoms and molecules do not obey the classical Newton equations of motion. People long ago tried to treat electronic motion classically, and found that features observed clearly in experimental measurements simply were not consistent with such a treatment. Attempts were made to supplement the classical equations with conditions that could be used to rationalize such observations. For example, early workers required that the angular momentum L=rx p be allowed to assume only integer mulitple of h/2(which is often abbreviated as h) which can be shown to be equivalent to the borh postulate n 2=2 Tr. However, unti scientists realized that a new set of laws, those of quantum mechanics, applied to light
17 this theory has proven to be reliable, but, of course, we are always searching for a “new and improved” theory that describes how small light particles move. If it helps you to be more accepting of quantum theory, I should point out that the quantum description of particles will reduce to the classical Newton description under certain circumstances. In particular, when treating heavy particles (e.g., macroscopic masses and even heavier atoms), it is often possible to use Newton dynamics. Briefly, we will discuss in more detail how the quantum and classical dynamics sometimes coincide (in which case one is free to use the simpler Newton dynamics). So, let us now move on to look at this strange Schrödinger equation that we have been digressing about for so long. I. The Schrödinger Equation and Its Components It has been well established that electrons moving in atoms and molecules do not obey the classical Newton equations of motion. People long ago tried to treat electronic motion classically, and found that features observed clearly in experimental measurements simply were not consistent with such a treatment. Attempts were made to supplement the classical equations with conditions that could be used to rationalize such observations. For example, early workers required that the angular momentum L = r x p be allowed to assume only integer mulitples of h/2p (which is often abbreviated as h), which can be shown to be equivalent to the Borh postulate n l = 2 pr. However, until scientists realized that a new set of laws, those of quantum mechanics, applied to light
microscopic particles, a wide gulf existed between laboratory observations of molecule level phenomena and the equations used to describe such behavior Quantum mechanics is cast in a language that is not familiar to most students of chemistry who are examining the subject for the first time. Its mathematical content and how it relates to experimental measurements both require a great deal of effort to master With these thoughts in mind, I have organized this material in a manner that first provides a brief introduction to the two primary constructs of quantum mechanics-operators and wave functions that obey a Schrodinger equation. Next, I demonstrate the application of these constructs to several chemically relevant model problems. By learning the solutions of the Schrodinger equation for a few model systems, the student can better appreciate the treatment of the fundamental postulates of quantum mechanics as well as their relation to experimental measurement for which the wave functions of the known model problems offer important interpretations rators Each physically measurable quantity has a corresponding operator. The eigenvalues of the operator tell the only values of the corresponding physical property that can be observed. Any experimentally measurable physical quantity F(e.g, energy, dipole moment orbital angular momentum, spin angular momentum, linear momentum, kinetic energy) has a classical mechanical expression in terms of the Cartesian positions( qil and momenta(pil of the particles that comprise the system of interest. Each such classical 18
18 microscopic particles, a wide gulf existed between laboratory observations of moleculelevel phenomena and the equations used to describe such behavior. Quantum mechanics is cast in a language that is not familiar to most students of chemistry who are examining the subject for the first time. Its mathematical content and how it relates to experimental measurements both require a great deal of effort to master. With these thoughts in mind, I have organized this material in a manner that first provides a brief introduction to the two primary constructs of quantum mechanics- operators and wave functions that obey a Schrödinger equation. Next, I demonstrate the application of these constructs to several chemically relevant model problems. By learning the solutions of the Schrödinger equation for a few model systems, the student can better appreciate the treatment of the fundamental postulates of quantum mechanics as well as their relation to experimental measurement for which the wave functions of the known model problems offer important interpretations. A. Operators Each physically measurable quantity has a corresponding operator. The eigenvalues of the operator tell the only values of the corresponding physical property that can be observed. Any experimentally measurable physical quantity F (e.g., energy, dipole moment, orbital angular momentum, spin angular momentum, linear momentum, kinetic energy) has a classical mechanical expression in terms of the Cartesian positions {qi} and momenta {pi} of the particles that comprise the system of interest. Each such classical
expression is assigned a corresponding quantum mechanical operator F formed by replacing the (pi) in the classical form by the differential operator-ihalaqj and leaving the coordinates q; that appear in F untouched. For example, the classical kinetic energy of n particles (with masses m) moving in a potential field containing both quadratic and linear coordinate-dependence can be written as F=∑1=1N(p2/2m+1/2k(qr-q0)2+L(q-q) The quantum mechanical operator associated with this F is F=∑|=1N(-h2/2ma2/0q12+1/2k(q-q)2+L(q-qp0) Such an operator would occur when, for example, one describes the sum of the kinetic energies of a collection of particles(the 2=1.N(pi2/2m)term), plus the sum of"Hookes' Law"parabolic potentials(the 1/2 2=1n k(ql-qI )2), and(the last term in F)the interactions of the particles with an externally applied field whose potential energy varies linearly as the particles move away from their equilibrium positions (qri Let us try more examples. The sum of the z-components of angular momenta (recall that vector angular momentum L is defined as L=rx p)of a collection of N particles has the following classical expression 1, N(PYj - yip
19 expression is assigned a corresponding quantum mechanical operator F formed by replacing the {pi} in the classical form by the differential operator -ih¶/¶qj and leaving the coordinates qj that appear in F untouched. For example, the classical kinetic energy of N particles (with masses ml ) moving in a potential field containing both quadratic and linear coordinate-dependence can be written as F=Sl=1,N (pl 2/2ml + 1/2 k(ql -ql 0) 2 + L(ql -ql 0)). The quantum mechanical operator associated with this F is F=Sl=1,N (- h2/2ml ¶ 2/¶ql 2 + 1/2 k(ql -ql 0) 2 + L(ql -ql 0)). Such an operator would occur when, for example, one describes the sum of the kinetic energies of a collection of particles (the Sl=1,N (pl 2/2ml ) term), plus the sum of "Hookes' Law" parabolic potentials (the 1/2 Sl=1,N k(ql -ql 0) 2), and (the last term in F) the interactions of the particles with an externally applied field whose potential energy varies linearly as the particles move away from their equilibrium positions {ql 0}. Let us try more examples. The sum of the z-components of angular momenta (recall that vector angular momentum L is defined as L = r x p) of a collection of N particles has the following classical expression F=Sj=1,N (xjpyj - yjpxj)
and the corresponding operator F=-1t 2i=I, N(Xjoldyj-yjaloxj) If one transforms these Cartesian coordinates and derivatives into polar coordinates, the above expression reduces to F=i>=1NO/0 The X-component of the dipole moment for a collection of N particles has a classical form of F Z for which the quantum operator is where Zie is the charge on the jth particle. Notice that in this case, classical and quantum forms are identical because F contains no momentum operators The mapping from f to F is straightforward only in terms of Cartesian coordinates. To map a classical function F, given in terms of curvilinear coordinates
20 and the corresponding operator is F=-ih Sj=1,N (xj¶/¶yj - yj¶/¶xj ). If one transforms these Cartesian coordinates and derivatives into polar coordinates, the above expression reduces to F = -i h Sj=1,N ¶/¶fj . The x-component of the dipole moment for a collection of N particles has a classical form of F=Sj=1,N Zjexj , for which the quantum operator is F=Sj=1,N Zjexj , where Zje is the charge on the jth particle. Notice that in this case, classical and quantum forms are identical because F contains no momentum operators. The mapping from F to F is straightforward only in terms of Cartesian coordinates. To map a classical function F, given in terms of curvilinear coordinates