function will be changed to become an eigenfuntion of the second operator and thus ne longer the eigenfunction of the first C. The Schrodinger equation his equation is an eigenvalue equation for the energy or Hamiltonian operator its eigenvalues provide the only allowed energy levels of the system I. The Time-Dependent equation If the Hamiltonian operator contains the time variable explicitly, one must solve the time-dependent Schrodinger equation Before moving deeper into understanding what quantum mechanics 'means, it is useful to learn how the wave functions Y are found by applying the basic equation of quantum mechanics, the Schrodinger equation, to a few exactly soluble model problems Knowing the solutions to these 'easy' yet chemically very relevant models will then facilitate learning more of the details about the structure of quantum mechanics The Schrodinger equation is a differential equation depending on time and on all of the spatial coordinates necessary to describe the system at hand(thirty-nine for the H2O example cited above). It is usually written H P=it ap/at
26 function will be changed to become an eigenfuntion of the second operator and thus no longer the eigenfunction of the first. C. The Schrödinger Equation This equation is an eigenvalue equation for the energy or Hamiltonian operator; its eigenvalues provide the only allowed energy levels of the system 1. The Time-Dependent Equation If the Hamiltonian operator contains the time variable explicitly, one must solve the time-dependent Schrödinger equation Before moving deeper into understanding what quantum mechanics 'means', it is useful to learn how the wave functions Y are found by applying the basic equation of quantum mechanics, the Schrödinger equation, to a few exactly soluble model problems. Knowing the solutions to these 'easy' yet chemically very relevant models will then facilitate learning more of the details about the structure of quantum mechanics. The Schrödinger equation is a differential equation depending on time and on all of the spatial coordinates necessary to describe the system at hand (thirty-nine for the H2O example cited above). It is usually written H Y = i h ¶Y/¶t
where y(qi, t) is the unknown wave function and h is the operator corresponding to the total energy of the system. This operator is called the hamiltonian and is formed, as stated above, by first writing down the classical mechanical expression for the total energy(kinetic plus potential) in Cartesian coordinates and momenta and then replacing all classical momenta p by their quantum mechanical operators pj--ihdlaqj For the h2o example used above the classical mechanical energy of all thirteen particles is E=2i( p2/2me +1/22jellrij-2a zaez/ri, a) + 2a(pa2/2ma 1/2 2b ZaZbe2/rab j where the indices i and i are used to label the ten electrons whose thirty cartesian coordinates are (qi and a and b label the three nuclei whose charges are denoted Zai and whose nine Cartesian coordinates are ai. The electron and nuclear masses are denoted me and mal, respectively. The corresponding Hamiltonian operator is ∑{-(2m)02/aq2+1/2e21j-alae2ia} ∑a{-(H2/2ma)a2/aqa2+1/2∑b Notice that H is a second order differential operator in the space of the thirty-nine
27 where Y(qj ,t) is the unknown wave function and H is the operator corresponding to the total energy of the system. This operator is called the Hamiltonian and is formed, as stated above, by first writing down the classical mechanical expression for the total energy (kinetic plus potential) in Cartesian coordinates and momenta and then replacing all classical momenta pj by their quantum mechanical operators pj = - ih¶/¶qj . For the H2O example used above, the classical mechanical energy of all thirteen particles is E = Si { pi 2/2me + 1/2 Sj e2/ri,j - Sa Zae 2/ri,a } + Sa {pa 2/2ma + 1/2 Sb ZaZbe 2/ra,b }, where the indices i and j are used to label the ten electrons whose thirty Cartesian coordinates are {qi} and a and b label the three nuclei whose charges are denoted {Za}, and whose nine Cartesian coordinates are {qa}. The electron and nuclear masses are denoted me and {ma}, respectively. The corresponding Hamiltonian operator is H = Si { - (h2/2me) ¶ 2/¶qi 2 + 1/2 Sj e2/ri,j - Sa Zae 2/ri,a } + Sa { - (h2/2ma) ¶ 2/¶qa 2+ 1/2 Sb ZaZbe 2/ra,b }. Notice that H is a second order differential operator in the space of the thirty-nine
Cartesian coordinates that describe the positions of the ten electrons and three nuclei. It is a second order operator because the momenta appear in the kinetic energy as pi and pa and the quantum mechanical operator for each momentum p=-ih a/aq is of first order The Schrodinger equation for the H2O example at hand then reads Xi(-(h2/2me)a210qi2+1/2 2e2/rij-2a Zae2/ri, a)Y +∑a{-(h2/2ma)a2/aqa2+12∑ b zazbe2/ab}平=iat The Hamiltonian in this case contains t nowhere. An example of a case where h does contain t occurs when the an oscillating electric field e cos(ot) along the x-axis interacts with the electrons and nuclei and a term 2aZie Xa E cos(ot)-2,ex, E cos(ot) is added to the hamiltonian Here. x and x denote the x coordinates of the a nucleus and the j electron, respectively 2. The Time-Independent equation I the Hamiltonian operator does not contain the time variable explicitly, one can solve the time-independent Schrodinger equation In cases where the classical energy, and hence the quantum Hamiltonian, do not
28 Cartesian coordinates that describe the positions of the ten electrons and three nuclei. It is a second order operator because the momenta appear in the kinetic energy as pj 2 and pa 2, and the quantum mechanical operator for each momentum p = -ih ¶/¶q is of first order. The Schrödinger equation for the H2O example at hand then reads Si { - (h2/2me) ¶ 2/¶qi 2 + 1/2 Sj e2/ri,j - Sa Zae 2/ri,a } Y + Sa { - (h2/2ma) ¶ 2/¶qa 2+ 1/2 Sb ZaZbe 2/ra,b } Y = i h ¶Y/¶t. The Hamiltonian in this case contains t nowhere. An example of a case where H does contain t occurs when the an oscillating electric field E cos(wt) along the x-axis interacts with the electrons and nuclei and a term Sa Zze Xa E cos(wt) - Sj e xj E cos(wt) is added to the Hamiltonian. Here, Xa and xj denote the x coordinates of the ath nucleus and the jth electron, respectively. 2. The Time-Independent Equation If the Hamiltonian operator does not contain the time variable explicitly, one can solve the time-independent Schrödinger equation In cases where the classical energy, and hence the quantum Hamiltonian, do not
contain terms that are explicitly time dependent(e. g, interactions with time varying external electric or magnetic fields would add to the above classical energy expression time dependent terms), the separations of variables techniques can be used to reduce the Schrodinger equation to a time-independent equation In such cases, H is not explicitly time dependent, so one can assume that p(qjs t)is of the form(n b, this step is an example of the use of the separations of variables method to solve a differential equation) Y(qj, t)=Y(q])F(t) Substituting this ' into the time-dependent Schrodinger equation gives (ai if aF/at=F(t)HY(qj) Dividing by y(qi f(t) then gives F1(i.F)=l(H平(q) Since F(t) is only a function of time t, and y(qi) is only a function of the spatial coordinates (qiB, and because the left hand and right hand sides must be equal for all values of t and of (qjb, both the left and right hand sides must equal a constant. If this constant is called e, the two equations that are embodied in this separated schrodinger equation read as follows
29 contain terms that are explicitly time dependent (e.g., interactions with time varying external electric or magnetic fields would add to the above classical energy expression time dependent terms), the separations of variables techniques can be used to reduce the Schrödinger equation to a time-independent equation. In such cases, H is not explicitly time dependent, so one can assume that Y(qj ,t) is of the form (n.b., this step is an example of the use of the separations of variables method to solve a differential equation) Y(qj ,t) = Y(qj ) F(t). Substituting this 'ansatz' into the time-dependent Schrödinger equation gives Y(qj ) i h ¶F/¶t = F(t) H Y(qj ) . Dividing by Y(qj ) F(t) then gives F-1 (i h ¶F/¶t) = Y-1 (H Y(qj ) ). Since F(t) is only a function of time t, and Y(qj ) is only a function of the spatial coordinates {qj}, and because the left hand and right hand sides must be equal for all values of t and of {qj}, both the left and right hand sides must equal a constant. If this constant is called E, the two equations that are embodied in this separated Schrödinger equation read as follows:
HY(qi=EY(qi) ih df(t dt=E F(t) The first of these equations is called the time- independent Schrodinger equation; it is a sO-called eigenvalue equation in which one is asked to find functions that yield a constant multiple of themselves when acted on by the Hamiltonian operator. Such functions are called eigenfunctions of H and the corresponding constants are called eigenvalues ofH For example, if H were of the form(-A2 /2M)a2/a02=H, then functions of the form exp(i mo )would be eigenfunctions because -H2/2Ma2/oφ2}exp(imp)={m22/2M}exp(imφ) In this case, m2 h2/2M is the eigenvalue. In this example, the Hamiltonian contains the quare of an angular momentum operator (recall earlier that we showed the z-component of angular momentum is to equal-it d/do) When the Schrodinger equation can be separated to generate a time-independent equation describing the spatial coordinate dependence of the wave function, the eigenvalue E must be returned to the equation determining f(t) to find the time dependent part of the wave function. By solving ih df(t/dt=E F(t)
30 H Y(qj ) = E Y(qj ), ih dF(t)/dt = E F(t). The first of these equations is called the time-independent Schrödinger equation; it is a so-called eigenvalue equation in which one is asked to find functions that yield a constant multiple of themselves when acted on by the Hamiltonian operator. Such functions are called eigenfunctions of H and the corresponding constants are called eigenvalues of H. For example, if H were of the form (- h2/2M) ¶ 2/¶f2 = H , then functions of the form exp(i mf) would be eigenfunctions because { - h2/2M ¶ 2/¶f2} exp(i mf) = { m2 h2 /2M } exp(i mf). In this case, m2 h2 /2M is the eigenvalue. In this example, the Hamiltonian contains the square of an angular momentum operator (recall earlier that we showed the z-component of angular momentum is to equal – i h d/df). When the Schrödinger equation can be separated to generate a time-independent equation describing the spatial coordinate dependence of the wave function, the eigenvalue E must be returned to the equation determining F(t) to find the time dependent part of the wave function. By solving ih dF(t)/dt = E F(t)