INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY SYMP. NO. 8, 413-420 (1974)

Conditional Probability Amplitude Analysis of Coupled Harmonic Oscillators

GEOFFREY HUNTER Department of Chemistry, York University, Toronto, Ontario, Canada

Abstract

The wave functions of two coupled harmonic oscillators are factorized as the product of a marginal probability amplitude and a conditional probability amplitude. This factorization exemplifies the general theory of conditional probability amplitudes in wave mechanics. Renormalization of the amplitudes leads to a very simple description of the motion of the coupled oscillators.

1. Introduction

The quantum mechanical problem of two coupled harmonic oscillators is solvable by transformation to normal coordinates [l]. In this paper the motion in terms of original coordinates will be used to exemplify the theory of conditional probability amplitudes in wave mechanics [2]. This is a nontrivial example of the theory in the sense that the two original coordinates are inseparable. However, it is sficiently simple that the partial integrals which arise can be evaluated analytically. Thus it is an excellent exemplary case, since the results are derivable without approximation. The resultant reduced Schrodinger equation for the motion along one of the original coordinates provides a new way of describing the motion of the coupled oscillators.

2. The Original Wave Equation

The original coordinates are designated by x and y. In general the respective reduced masses px and p, and the force constants K , and K , will be different. For the sake of simplicity in the subsequent analysis, the general case is restricted to the one where these four parameters satisfy the relation

(1) KxPy = KyPx For this case the Hamiltonian H can be written in reduced atomic units in the form [l]

An eigenfunction of H is designated by $(x, y), and with A as the energy eigenvalue in reduced atomic units, the original Schrodinger equation is

(3) H$ = A$

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The parameter a may take any value in the range 0 5 a < 1. The case a = 0 corresponds to uncoupled oscillators, and the strength of the coupling interaction increases with the magnitude of a. For a # 0 the variables x and y are not separable in the differential equation (3).

3. The Normal Coordinate Solution

The normal coordinates 5 and [ are defined by [ 13

(4) 5 = f ( x + Y ) , i = t ( x - Y )

Transformation of Equation (3) into these coordinates eliminates the cross term in the potential energy, so that 5 and [ are separable. The wave function $(x, y ) may be written as a product, and the eigenvalue 1 as a sum:

(5) Further transformation to reduced normal coordinates w and z defined by

(6) simplifies each of the separated wave equations to

$(x, y ) = Z(5) Z ( [ ) ,

w = (1 + a)'/45,

1 = 2 + A6

z = (1 - a)'/4[

d2Q dq2 - + q2Q = 1'Q

where q = w or z , and Q(q) = Z(5) or Z ( [ ) .

cillator [l, sec. 111-111. Thus the eigenfunctions Q(q) and eigenvalues Aq are Equation (7) is the well-known equation for a one-dimensional harmonic os-

(8) Qn (4) = e- 1/2qZHn (4)

1; = 2n + 1

where the quantum number n is any nonnegative integer (0,1,2, ...). H,(q) is the nth-order Hermite polynomial [l, sec. 111-1 13.

(9) $nm = e- 1/2acZHn( &() e- 1/2b62Hm( f i t )

(10) I f lm = a(2n + 1) + b(2m + 1)

The solutions are characterized by the two quantum numbers n and m (both any nonnegative integer). The parameters a and b are related to o! by

(11) a = (1 + b = (1 - a)'/'

a2 - b2 = 2a

Transformation from w and z back to 5 and [ yields the results

a' + b2 = 2,

A noteworthy feature of the solution is that although the energy Anm is the sum of two harmonic oscillator energies, these oscillators in general have different funda- mental frequencies. Except for special values of a, all the energy levels defined by Equation (10) are nondegenerate.

COUPLED HARMONIC OSCILLATORS 41 5

For small values of the coupling constant a the eigenvalue spectrum is better expressed in terms of two quantum numbers k andjrather than by n and m of Equation (10). The relationship between these alternative quantum numbers is

(12) k = n + m , 2 j = n - m

The eigenvalue spectrum in terms of k and j is

(13) Ak' = ( k + l)(a + b) + 2 j ( a - b)

k takes any nonnegative integral value, a n d j takes the range of values from - k / 2 to + k / 2 in integer steps. Thus j is integral if k is even, and half-integral if k is odd. To first order in a, a + b = 2 and a - b = a, so that the principal quantum number k largely determines the separation of the energy levels. The fine structure within the group of k + 1 levels for a particular value of k is determined by the term 2j(a - b). The original normal coordinate quantum numbers n and m will be used to designate the wave functions.

4. Normalization of @,,,

The normalized wave function corresponding to t,bnm is denoted by Y',,, so that they are related by

(14) y n m = Nnmt,bnm

where [l, sec. 111-111

5. Factorization of Ynm

The total wave function Ynm(x, y) may be factorized into a marginal probability amplitude fnm(y) and a conditional probability amplitude 4nm(xJy) :

(16) y n m (x, Y) = f m ( ~ ) 4nrn(XlY)

The alternative factorization f (x) 4 (ylx) need not be considered explicitly, since the Hamiltonian (2) is symmetric with respect to interchange of x and y. For this reason the eigenfunctions Y,,(x, y) will either be symmetric or antisymmetric with respect to interchange of x and y. For the symmetric eigenfunctions the alternative factorization is identical with the original one [Equation (16)]. For the antisymmetric Y functions the two factorizations are simply related by replacement of x and y by - x and - y.

The marginal density f;, is defined by *+a,

f i m = J [Ynrn(X, y)12dX - m

(17)

This defines the marginal amplitudef,,(y) except for a phase factor, which will be

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taken to be unity. The conditional amplitude &,, (xIy) may then be obtained by divi- sion of Y n m ( x , y ) by fnm(y) [Equation (16)].

6. The Reduced Wave Equation

Substitution of the product form (16) into Equation (3) followed by averaging over the space spanned by x , yields the following reduced Schrodinger equation satisfied by the marginal amplitude f , , ( y ) [ 2 ] :

The potential energy in this Schrodinger equation is the expectation value:

It is especially noteworthy that the operator H in Equation (19) is the complete original Hamiltonian of the coupled oscillators defined in Equation (2). The deriva- tion of & , , ( x l y ) in Section 5 implies the normalization

(20) J-Y C4,,,, ,(XlY)l2dX = 1

for all values of y . Although U n m ( y ) is formally defined by Equation (19), it may also be obtained by

substituting the known form of f,,,,,(y) and A,,,,, into Equation (18). This is a simpler procedure, since the only partial integration involved is that in Equation [17]. The partial integrations may be performed analytically [3].

It is also noteworthy that although Equation (18) is a differential equation satisfied by fn,(y) , the other proper solutions of this differential equationf (which certainly exist as shown below) do not necessarily make the product f ( y ) Cp,(xlyj an eigen- function of H .

I

7. The Ground State

From Equations (5)-(9), (14), and (15), Yoo in terms of the original coordinates x and y is

2 X 2 Y 2

(21) Yoo = ( $)1’4 exp( - - (a + b) - -(a 4 + b) - 4 Evaluation of the integral in Equation (17) for this case produces foo and q500

COUPLED HARMONIC OSCILLATORS 417

The potential Uoo in the reduced Schrodinger Equation (18) is

u,, = -

This potential is a parabolic harmonic oscillator potential with minimum energy at y = 0 of 2/(a + b). f o o ( y ) is the zeroth-order Hermite function corresponding to the potential Uoo. Although the harmonic oscillator with potential Uoo has a set of eigenvalues 2 [ 1 + ab(2k + l ) ] / ( a + b), the only one which is an eigenvalue of H is the zeroth-order one ( k = 0). This exemplifies the inseparability of the eigenvalue spectrum noted at the end of Section 6.

The conditional amplitude 4,, has the same functional form as Yo, : c$oo differs from Yo, only by the coefficient of y 2 in the exponential. This coefficient is typically smaller in 4oo than in Yo,. To first order in a, a = 1 + 4 2 , b = 1 - 4 2 , so that a + b = 2 and a - b = a. Thus to first order the coefficient of y 2 in Yoo is a + b = 2 compared with (a - b)2/(a + b) = a2 /2 in $,,. The effect of this smaller exponent is that +,, is a more slowly varying function of y than is Yoo.

8. Excited States

The excited state total wave functions Y,,, have the general form

(25) y n m = G n m ~ o o Gn, is a polynomial in x and y of order n + rn = k. It is an odd or even polynomial corresponding to odd and even values of k. The Gaussian exponential factor of all the wave functions is the same as that of Yo, [Equation (21)]. The first five excited states (k = 1 and 2) have the following polynomial factors G,,,:

I Go2 = - -(x - y)' - 1 f l 2 i"

TheorderoftheeigenvaluespectrumofthesestatesisA,, < Aol < A l 0 < A,, c A l l < A,, for small values of a.

Determination of the marginal amplitudes f,, for the excited states is accom- plished by partial integration of Y&, in Equation (17). This produces a polynomial Fn, multiplied by the Gaussian exponential form of f:, [Equation (22)l . Thus,

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(27) f n m = F2’fOo Combining Equations (16), (25), and (27), the excited conditional amplitude &, is given by

F,, is an even polynomial in y of order 2(n + m) = 2k. The polynomial F,, is related (28) d n m = G n m F A 1 ’ 2 ~ o o

1 to Grim by (29) F,, = ~ G&Ygo dx = J-: Gim$go dx

f g o J-a

For the first five excited states F,, has the explicit form

F1l = 4(a + b)

2 4 a + h a + b a + b F02 = A {1 - F, , + ?[ F , , + (”-”) (->’ 4ayZ + a (A>’]} F z o = Foz(b, a, Y)

The meaning of the notation F , , = Fo,(b, a, y) is that F , , is obtained from Fol by interchanging a and b; similarly for F z o = Foz(b, a, y).

The potential U,, occuring in the reduced Schrodinger equation may be obtained by substituting $,,from Equation (28) into Equation (19), or equivalently, and more simply, by substituting fn, from Equation (27) into Equation (18) together with the known value of A,,,,. The general result has the form.

2aby dlogF,, a + h dy

U,, = Uoo + 2(an + bm) - -

1 dlogF, 1 d’logF,, +4( dy ) + 5 dy2

Thus the potential U,, is only a harmonic oscillator potential for the ground state (n = m = 0; Foo = 1). For the excited states the harmonic potential Uoo [Equation (24)] is modified by the terms involving logF,,.

9. Renormalized Excited Amplitudes

The excited state reduced potential U,,(y) [Equation (31)] is rather a complicated function of y by comparison with Uoo [Equation (24)]. The complicating terms (those involving derivatives of IogF,,) arise essentially because of the factor F,!L2 in j;, [Equation (27)].fn,(y) is the appropriate solution of the reduced Schrodinger Equation

COUPLED HARMONIC OSCILLATORS 419

(18) corresponding to the eigenvalue An, :

(32) I,, = Loo + 2(an + bm)

Thus the potential U,, has the remarkable property that if the terms involving derivatives of log F,, are removed, the lowest eigenfunction of Equation (18) cor- responds to precisely the same eigenvalue I , [Equation (32)]. In this case the ap- propriate solution of Equation (18) is simply foo. The constant term 2(an + bm) in U,, is just the amount of energy by which A., is above Loo [Equation (32)].

This observation indicates that a much simpler factorization of excited state wave functions results from renormalization of fnm and +,, by removal of F,, from Equations (27) and (28). Thus the renormalized amplitudesf’,, and &,,, are defined by

(33) y n m = L m 4 n m = f b r n d m

4;m = F 2 ’ 4 n m = Gnrn400

f n m = F2’Lm = f o o

These renormalized amplitudes retain the essential character of the original ampli- tudes. The potential resulting from the renormalized &,,, is denoted by U;,:

’+ m

u,, = - J &nrnH4;mdx F n m - m

(34)

Because #,, no longer satisfies the normalization condition (20), the reduced Schro- dinger equation contains an additional term involving df’,,,Jdy :

d‘f’nm d log F n m dYm dY2 dY dY

+ u n m f ’ n m = L n f ’ n r n (35)

By virtue of Equation (33) (f’,, = f o o ) and Equation (22),

- 2aby dY a + b f b m

-- -- dfbm

Hence Equation (35) becomes

(36)

where the effective scalar potential Uim is

(37) 2aby dlogF, a + b dy

u;, = u,, + -

= Uoo + 2(an + bm)

The new reduced Schrodinger Equation (36) provides a remarkably simple picture of the different states of motion of the coupled oscillators. The motion along the coordinate y is that of a one-dimensional harmonic oscillator, which is always in’

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its zero-point vibrational state (foe (y)). The different states of motion are associated with the different potentials ULm [Equation (37)]. These potentials are all harmonic with the same force constant; they differ only in being displaced upward from the ground state potential by the constant amounts 2(an + bm). This oscillator has a dflerent force constant 2ah/(a + b) from that apparent in the original Hamiltonian (2), and from that in the normal coordinates (a or h ) [Equation (7)].

Bibliography

[l] L. Pauling and E. B. Wilson, Introduction to Quantum Mechanics (McGraw-Hill, New York, 1935),

[2] G . Hunter, Conditional Probability Amplitudes in Wave Mechanics (to be published, 1974). [3] I. S . Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products (trans]. by A. Jeffrey)

sec. XI-41.

(Academic Press, New York, 1965), sec. 3.323.