New Potentials for Old: The Darboux Transformation in Quantum

Research: Science and Education

New Potentials for Old: The Darboux Transformation in Quantum Mechanics Brian Wesley Williams* Department of Chemistry, Bucknell University, Lewisburg, PA 17837; *[email protected] Tevye C. Celius Department of Chemistry and Biochemistry, Ohio Northern University, Ada, OH 45810

In elementary quantum mechanics, the number of solvable systems whose wavefunctions can be exactly represented in terms of basic mathematical functions is relatively small. Luckily for chemists, many of these solvable systems, such as the particlein-a-box, the hydrogen atom, and the harmonic oscillator, serve as good starting points for understanding real atoms and molecules. Nevertheless, there remains continuing interest in seeking improved models applicable to physical and chemical systems and the more fundamental question of whether other exactly solvable quantum mechanical systems can be found. An earlier article in this Journal explored an approach to finding novel, exactly solvable quantum mechanical systems (1). This method depended upon knowing the solutions to secondorder ordinary differential equations. A different approach to this problem is examined here: the Darboux transformation. Like the method cited, the Darboux transformation also depends upon knowing the solutions to second-order ordinary differential equations. In this approach, the differential equation is of a very particular type: namely, the Schrödinger equation itself ! Essentially, the Darboux transformation offers a method whereby knowing one solvable quantum mechanical system’s potential, wavefunctions and energies allows the mathematical generation of a second solvable potential and its associated wavefunctions and energies. The Darboux transformation is also closely connected to modern operator-oriented approaches to quantum mechanics, including the factorization method (2) and supersymmetric quantum mechanics (3). Many physical and theoretical chemists do not appear to know of the Darboux method or at least of the novel solvable potentials closely associated with the simple systems mentioned above. In the following, we sketch the mathematical basis of the Darboux transformation and illustrate its use by applying it to the one-dimensional particle-in-a-box and harmonic oscillator potentials to generate some related exactly solvable systems. To conclude, we also briefly indicate how this transformation is connected to the factorization method and supersymmetric quantum mechanics. Many earlier articles (including refs 4 and 5) have already discussed the potentials and ideas we present here. Despite this, we believe there is value in offering a detailed, elementary exposition on the Darboux transformation accessible to individuals comfortable with single-variable calculus. The Darboux Transformation The basic one-dimensional Schrödinger equation

576

2 h d 2 : x

V x E : x 0 2m dx 2

takes the form

d 2 : x

u x e : x 0

dx 2

(1)

with the definitions 2m

u x

h

2

V x

and

e

2m h

2

E

Now suppose there are known solutions to a differential equation in this form. Pick a particular e1 and Ψ1(x) that are suitably well-behaved such that

d2 :1 x

dx

2

:1 x

(2)

u x e1

and a function σ(x) can be defined as d :1 x

T x

(3)

dx

:1 x

Taking the derivative of σ(x) using the quotient rule gives

d T x

dx

d2 :1 x

dx 2

:1 x

d :1 x

dx

2

:1 x

u x e1

T 2 x

T 2 x

u x e 1

d T x

dx

(4)

Now, define a new function φ(x) in terms of Ψ(x) and σ(x) by the expression d : x

K x T x : x

(5) dx

Journal of Chemical Education • Vol. 85 No. 4 April 2008 • www.JCE.DivCHED.org • © Division of Chemical Education


mixing the general Ψ(x) wavefunctions having energies e with the specific Ψ1(x) through σ(x). Taking the derivative of the new function φ(x) with respect to x gives d : x

d K x

d 2 : x

dT x

: x T x

2 dx dx dx dx u x e : x u x e1 : x

T 2 x : x T x

e1 e : x T x t T x : x

d : x

dx

(6)

d: x

dx

Taking the second derivative of φ(x) starting from this result gives d2 K x

dx 2 e1 e

d : x

dK x

d T x

K x

T x

dx dxx dx

e1 e K x T 2 x K x

u x 2

d T x

e K x

dx

d 2 K x

dx

2

(7)

d :1 x

dx :1 x

d 2 K1 x

2

(9)

1

:1 x

T x K1 x

(10)

d K 1 x

d T x

K1 x

dx dx d T x

T 2 x K1 x K1 x

dx T x

d T x

K x

dx

d T x

© ¸ ªu x 2 e ¹ K x 0 dx « º

(8)

with a new potential v x u x 2

d K1 x

dx

dx

and the result that φ(x) satisfies a differential equation of Schrödinger form

:1 x

and

d T x

K x

dx

e1 e K x u x e1 K x 2

1

K1 x

leads to

d : x

e1 e

e1 e T x : x

dx d T x

K x

T 2 x K x dx

tential has no solution for energy e1. The first difficulty can be overcome by recognizing that while proper square integrable wavefunctions must indeed approach zero at the boundaries of the space in which they exist, if Ψ1(x) is chosen such that it lacks nodes in the region between the boundaries, the function φ(x) will not become infinite between these boundaries. Since the ground states of solvable potentials lack nodes, these same ground states can serve as the Ψ1(x) used to define new φ(x) solving the new potential. Specific examples of this type of transformation will be given below. An alternative definition for φ1(x) may permit recovery of a solution to the new potential at e1. The definition

e1 e : x T x K x

Ψ1(x). Second, applying the definition given for φ(x) when Ψ(x) = Ψ1(x) results in φ1(x) = 0, suggesting that the new po-

dT x

dx

defined in terms of the old potential and the derivative of the function σ(x). This new potential has the same energies e at each new φ(x) as the old potential did for each Ψ(x). Because σ(x) is also determined from the original differential equation through the choice of a particular solution Ψ1(x), knowing the solutions to one solvable Schrödinger equation may permit construction of a second. Two difficulties remain at this point. First, the scheme outlined here depends on the choice of a “suitably well-behaved”

u x 2

(11)

d T x

e1 K1 x

dx

having the same form as the differential equation for φ(x) and the new potential. A problem with this alternative definition arises if Ψ1(x) is itself a square integrable solution to the original potential, since inverting it then leads to unbounded values at the boundaries. If the ground-state wavefunction for a potential were chosen as Ψ1(x), for example, while its inverse certainly solves the new potential with energy e1, the inverse cannot be a proper wavefunction for the new potential since it is not square integrable. The practical consequence of choosing the ground state of the original potential as Ψ1(x) is to permit generation of a new potential that has the same energies as the original potential, except for the ground state. These considerations allow for another possibility. If a Ψ1(x) solving the initial differential equation can be found such that its inverse does meet the boundary conditions and remains finite between the boundaries, it then becomes a viable, square integrable solution for the new potential with energy e1. This solution might not be a square integrable solution for the original potential! To put things another way, while unbounded solutions to a solvable potential certainly cannot be proper wavefunctions for that potential, if they are nodeless, their inverses can be used as the basis for generation of a new solvable potential. The new solvable potential has all the energies of the old potential and in addition includes a solution with the energy e1. Examples of this type of transformation will also be given below.

© Division of Chemical Education • www.JCE.DivCHED.org • Vol. 85 No. 4 April 2008 • Journal of Chemical Education

577


Application of the Darboux Transformation

The discussion above would perhaps suggest 1 csc x

sin( x )

Below, the Darboux transformation will be applied to some well-known solvable potentials to generate novel, exactly solvable potentials. Initially, the one-dimensional particle-in-a-box case will be used to derive a potential having exactly the same energies except for the ground state. The “cosecant squared” potential derived this way turns out to be a particular case of a more general Pöschl–Teller potential (e.g., ref 6). This more general case is then investigated, demonstrating how a new potential possessing all the energy states of this original potential plus an additional state can be generated. Finally, the harmonic oscillator potential is approached in this fashion, with derivation of a new potential having all of the harmonic oscillator energy levels plus an additional new ground state.

The Particle-in-a-Box

Taking the first and second derivatives of φn(x),

Consider the one-dimensional particle-in-a-box potential (7) in the form

d 2 : x

dx

2

2m 2 E : x e : x

h

(12)

where the boundaries of the potential are assumed for convenience to lie at x = 0 and x = π. The unnormalized wavefunctions are then given as Ψ(x) = sin(nx) with energies

E

but since this function becomes unbounded at x = 0 and x = π, it is not square integrable and so cannot represent a proper wavefunction. If the Darboux procedure is correct, the new unnormalized wavefunctions φn(x) should now represent solutions to the differential equation

e

T x

2m n h 2 2m h

d sin x

dx sin x

n

578

n cos nx 2 csc 2 x cot x sin nx

n csc2 x coos nx n csc 2 x cos nx

n2

2, 3,, 4, …

n2 2 csc 2 x K n x

shows this to be true. These wavefunctions also meet the boundary conditions φn(x) = 0 at x = 0 and x = π. This can be demonstrated as follows. The second term in the wavefunction φn(x),

cot x

d sin nx

cot x sin nx

dx n cos nx cot x sin nx

(15)

2 csc 2 x n cos nx cot x sin nx

cot x sin nx

cos x

sin nx

sin x

appears indeterminant as written at x = 0 and x = π since both the sin(nx) and sin(x) terms approach zero. Applying L’Hôpital’s rule shows that at the boundaries the ratio becomes

d T x

csc 2 x

dx Kn x

(14)

3

n 2 n cos nx cot x sin nx

and so based on eq 5, wavefunctions

d x2

2 csc 2 x K n n2 K n

n 2 cot x sin nx

and

d 2 Kn x

dx2

n h 2m 2 2

d 2 Kn

d Kn x

n 2 sin nx csc 2 x sin nx

dx n cot x cos nx

2 2

where n = 1, 2, 3, ... . The ground-state wavefunction sin(x) has no nodes between x = 0 and x = π, making it a suitable choice for Ψ1(x). Applying eq 3 of the Darboux transformation gives

and

making

K1 ( x )

(13)

lim

x n 0, Q

sin n x

sin x

n cos nx

cos x

x 0, Q

For the whole second term, the denominator here is then cancelled out by the leading ‒cos(x) term, and the second term in φn(x) then becomes exactly equal and opposite to the leading ncos(nx) term.



The potential 2csc2(x) is shown in Figure 1 and has the same energies e as the original particle-in-a-box potential except for the ground state. In principle, it should now also be possible to apply the Darboux transformation to the ground state φ2(x) [proportional to sin2(x)]. In practice, this only results in a simple multiple of the same potential and so has limited interest. This “cosecant squared” potential is really only a particular example of the more general Pöschl–Teller class. In the next section, the Darboux transformation will be applied to a more general case, showing how energy states can be added to known solvable potentials to give new solvable potentials.

14

Energy /

h2 2m

12

K3 ä cos(x) sin2(x)

10 8 6

K2 ä sin2(x)

4 2 0

d2 : x

dx

2

n

B : x

2

2.0

2.5

3.0

(16)

has solutions Ψ(x) = sinα(x)Cnα(cos(x)) (n = 0, 1, 2, ...) where the Cnα(cos(x)) are the orthogonal Gegenbauer (ultraspherical) polynomials expanded in terms of cos(x) (8). This differential equation has exactly the form of a Schrödinger equation as transformed in eq 1 for a generalized “cosecant squared” potential u(x) = α(α − 1)csc2(x). When α = 1, the form for a particlein-a-box is obtained; when α = 2, the form for the 2csc2(x) potential derived above. Interestingly, for α = 1, the solutions Ψ(x) = sinα(x)Cnα(cos(x)) simplify to the form Ψ(x) = sin[(n + 1)x] expected (9). This generalized “cosecant squared” potential serves as a starting point for demonstrating how the Darboux transformation process can permit generation of a new potential having all of the energy levels of the original potential plus an additional energy level. The approach here depends upon finding a solution to an initial Schrödinger differential equation that, while not a valid wavefunction itself, still serves to generate a valid wavefunction if it is nodeless and its inverse meets any necessary boundary conditions. For example, start by considering solutions to the 2csc2(x) potential. From eq 16, when α = 2, Ψ(x) = sin2(x)Cn2(cos(x)) with energies e = (n + 2)2 for (n = 0, 1, 2, ...). The lowest energy allowed is e = 4, and these wavefunctions clearly meet the boundary conditions Ψ(0) = 0 and Ψ(π) = 0 through the sin2(x) factor. However, for α = ‒1, u(x) also equals 2csc2(x), since u(x) = α(α − 1)csc2(x). The general solution Ψ(x) = sinα(x)Cnα(cos(x)) suggests Ψ(x) = sin‒1(x) = csc(x) as a possible solution for α = ‒1 and n = 0, giving e = 1. Differentiation shows this is true: d d d csc x cot x

csc x dx dx dx csc x cot 2 x

csc 2 x csc x

csc x csc 2 x 1 csc 2 x

cs c x 2 csc 2 x 1

1.5

Figure 1. The potential 2csc2(x). The first two energies and the unnormalized wavefunctions φ2(x) and φ3(x) (from eq 13) are shown.

B B 1 csc 2 x : x

1.0

x

The differential equation

0.5

0.0

The General “Cosecant Squared” Potential

(17)

Now, while this solution solves the Schrödinger differential equation for the 2csc2(x) potential, it cannot be a wavefunction because it does not meet the boundary conditions at x = 0 and x = π. Nevertheless, because it is nodeless, it can still serve as a particular solution Ψ1(x) and so act as the starting point for a Darboux transformation for the original potential. Followng eq 3 gives

T x

csc x cot x cot x

csc x

(18)

and ultimately a potential through eq 11 given as

v x u x 2

d T x

dx

2 csc 2 x 2 csc 2 x

(19) 0

with the energies (n + 2)2 for n = 0, 1, 2, ... associated with the original 2csc2(x) potential. In other words, the particle-in-a-box potential has been recovered from the 2csc2(x) potential, with all of its original energies except for e = 1! From eq 9, we also know that

Ke 1 x

1 1 sin x

:e 1 x

csc x

is a solution, and since it also meets the boundary conditions, we can add this energy level e = 1 to the energies (n + 2)2 for n = 0, 1, 2, ... . The “new” potential has all the same energy levels as the starting potential plus an additional energy that the starting potential did not have. This approach is most interesting when it is used to generate potentials with novel forms. For the generalized “cosecant squared” potential u(x) = α(α − 1)csc2(x), because the term α(α − 1) is quadratic, for positive α values giving cosecant squared type potentials, there are also negative α values giving the same potential, as demonstrated above. Possible wavefunctions Ψ for these negative α are problematic in that they were defined above in terms of a leading sinα(x) = csc|α|(x) factor and an orthogonal Gegenbauer polynomial formally defined only for α > ‒1/2 (10).


579


However, an identity between the Gegenbauer polynomials and the hypergeometric functions 2F1(a, b, c, x2) allows some nodeless solutions. First, note that the hypergeometric functions 2 2F1(a, b, c, x ) are defined by

2 2 F1 a , b, c , x

ab x2 c 1!

1

a a 1 a 2 t b b 1 b 2 x 6 c c 1 c 2

3!

(20)

1

1 1

1 / 2

x2 1!

1 0 1 0

1 / 2 3 / 2

x4 2!

1 0 1 1 0 1

1 / 2 3 /2 5 / 2

d : x 2 cot x csc2 x 1 2 cos 2 x 4 cot x

2 d : x 2 csc 4 x 1 2 cos 2 x

dx 2 4 cot 2 x csc 2 x 1 2 cos 2 x

8 cot 2 x 4 csc 2 x

x6 3!

(22)

showing this is true. This function (where α = ‒2) is also a solution to the Schrödinger differential equation associated with the potential u(x) = α(α − 1)csc2(x) for α = 3 and e = 0. However, it cannot be a correct wavefunction for this potential, since it does not meet the boundary conditions. Only the wavefunctions Ψn(x) = sin3(x)Cn3(cos(x)) and energies (n + 3)2 (n = 0, 1, 2, ...) would be permitted. But consider what happens upon applying the Darboux transformation to the α = ‒2 solution Ψ(x) = csc2(x)[1 + 2cos2(x)]. This gives

T x

d: x

dx

(26)

: x

2 cot x csc 2 x 1 2 cos 2 x

1 2x2

(23)

which for the proper choices of n and negative α are not themselves valid wavefunctions, but which are nodeless and also meet the boundary conditions. These are exactly the kinds of solutions needed to perform the Darboux transformation generating a new potential adding an energy level to those for original potentials having positive α.

4 cot x sin2 x

which has no nodes for any real x. Applying this to the generalized “cosecant squared” potential means that a variety of solutions to Schrödinger differential equations with u(x) = α(α − 1)csc2(x) can be found with the general form

580

(25)

6 csc 2 x : x

…

: x csc B x F n, n B, 1 / 2, cos 2 x

2 1

2 csc 2 x 4 cot 2 x 4 t 1 2 cos 2 x csc 2 x

(21)

This relationship permits an operational definition for α (x) when α is negative. Perhaps even more interesting is that C2n for the right choice of positive n and negative α, polynomials can be generated that have no negative coefficients associated with different powers of x2. For example, suppose n = 1 and α = ‒2. The definition for 2F1(‒1, ‒1, 1/2, x2) gives 2 2 F 1 1, 1, 1 / 2, x

(24)

and

(11) where c cannot be a negative integer. Now, if either a or b is a negative integer, the sum in eq 20 terminates and 2F1(a, b, c, x2) becomes a polynomial. Gegenbauer polynomials with even numbered polynomial indices turn out to be proportional to hypergeometric functions 2F1(a, b, c, x2) with particular parameters a, b, and c (12):

: x csc 2 x 2 F1 1, 1, 1 / 2, cos 2 x

csc 2 x 1 2 cos 2 x

dx

…

C 2Bn x u 2 F1 n, n B, 1 / 2, x 2

is obtained as a possible solution to the Schrödinger differential equation for u(x) = α(α − 1)csc2(x) = 6 csc2(x) with energy e = 0. Differentiation gives

a a 1 b b 1 x 4 c c 1

2!

A specific example should help to clarify things. For n = 2 and α = ‒2 in eq 16, and the definition given in eq 22, the wavefunction

1 2 cos 2 x

2 cot x 4 cot x cos 2 x sin 2 x

6 cot x

2 cos 2 x

1 2 cos 2 x

and

d T x

6 csc 2 x 12 cot x sin 2 x

2 dx 2 cos 2 x

2 cos 2 x

(27)


Research: Science and Education 20

leading to the Darboux “partner” potential 2 v x 6 csc 2 x 12 csc x 24 cot x sin 2 x (28) 2 2 cos 2 x

2 cos 2 x

h2 2m

10

K0 x

sin 2 x

dx

2

x 2 : x 2n 1 : x

(29)

0.0

d 2 : i x

d ix

d : ix

dx 2

(30)

H6 ( x ) 64 x 6 480 x 4 720 x 2 120 ... Generally, these Hermite polynomials H2n(x) have the property that they can be represented as polynomials in x2 with alternating terms. Multiplication by the factor exp(‒x2/2) then gives the complete wavefunction Ψ2n(x) = exp(‒x2/2) H2n(x). Consider changing the variable x in eq 29 to another variable, say y, that is a function of x. Since y is a function of x, the second derivative term (which is now with respect to y) can be related to terms involving derivatives of Ψ with respect to x using the chain rule:

d 2 : y

d x2

d 2 : y

d : y d 2 y d y d x2 dy2

dy dx

2

(31)

Now suppose y = ix, that is, just x multiplied by the imaginary value i = (‒1)1/2. Since i is a constant, dy d2y i and 0 dx dx2

giving

d 2 : ix

dx2

d 2 : ix

d ix

2

i 2

d 2 : ix

d ix

2

2

2

so finally

H 4 ( x ) 16 x 4 48 x 2 12

1.5

2.0

2.5

3.0

Applying all this to eq 29 gives

where the Hn(x) are the Hermite polynomials (13). Initially, consider just those Hermite polynomials where the polynomial index is even (e.g. n = 0, 2, 4, 6, ...):

H2 ( x ) 4 x 2 2

1.0

Figure 2. A general “cosecant squared” potential u(x) and a related Darboux transform partner potential v(x). An additional energy state e = 0 has been added in the partner potential to the original e = 9 and e = 16 states.

0.5

x

:n x exp x 2/ 2 Hn x n 0, 1, 2, ...

H0 ( x ) 1

12 csc 2(x) 24cot(x) sin(2x) á 2 á cos(2x) [2 á cos(2x)] 2

20

corresponding to the harmonic oscillator potential u(x) = x2 has as its solutions

u(x) ä 6csc 2(x)

10

1 2 cos 2 x

Harmonic Oscillator The differential equation d2 : x

0

v(x) ä 6csc 2(x) ź

meeting the boundary conditions. The potentials u(x) = 6csc2(x) and v(x) are compared in Figure 2.

Energy /

that has the same energies (n + 3)2 (n = 0, 1, 2, ...) as the original potential u(x) = 6csc2(x) and the additional energy e = 0 and wavefunction

d 2 : i x

dx 2

ix : ix 2 n 1 : ix

2

ix : ix 2 n 1 : ix

2

x 2 : i x 2n 1 : i x

(32)

Multiplying this result by ‒1 shows that the Ψn(ix) represent solutions to the Schrödinger equation for the harmonic oscillator for negative energies e = ‒(2n + 1). Explicit solutions for Ψ2(ix), Ψ4(ix), Ψ6(ix), ... then take the forms

:2 i x exp x 2/ 2 4 x2 2

:4 i x exp x 2/ 2 16 x 4 48 x2 12

:6 i x exp x 2/ 2 64 x 6 480 x 4 720 x 2 120

…

(33)

and so on. Substitution of ix for x has had the effect of ensuring that the coefficients of the polynomials for the H2n(ix) all have the same sign. These solutions are not acceptable wavefunctions because they do not meet the boundary conditions Ψ2n(ix) = 0 at x = ±∞. However, they do represent solutions to the differential equation for the harmonic oscillator that do not possess nodes. So, these solutions can serve as the starting functions needed to perform a Darboux transformation on the harmonic oscillator potential! Further, since the inverse of these solutions do meet the boundary condition that [1/Ψ2n(ix)] = 0 at x = ±∞, through eqs 8–10 the energy associated with the particular solution used for the transformation will also be associated with the new potential. Choosing one of these functions as the basis for a Darboux transformation has the effect of adding an energy level


581


to those originally associated with the harmonic oscillator. This is similar to the situation shown above for the transformation of “cosecant squared” type potentials. As an example, consider the wavefunction Ψ2(ix). Differentiation gives d: 2 i x x exp x 2/ 2 4 x 2 2 exp x 2/ 2 8 x

dx and d2 : i x 2

d x2

exp x 2/ 2 4 x 2 2

8 x 2 exp x 2/ 2 8 x 2 exp x 2/ 2

8 exp x 2/ 2

x

(34)

exp x / 2 4 x 2

2

2

1 4 exp x 2/ 2 4 x 2 2

x :2 ix 5 :2 ix

so that Ψ2(ix) represents a solution to the harmonic oscillator differential equation for e = ‒5. Following eq 3 gives

T x

d :2 ix dx :2 i x

(35)

4x

x

2x

2

1

d2 dx2

and H :1 x

8 2 x2 1

16 x

2 x

2

32 x 2

2x

2

1

1

K5 x

1

:2 i x

(36)

4x

2

This novel potential is compared to the original harmonic oscillator potential in Figure 3. The overall effect has been to find a new potential having all of the energies of the harmonic oscillator plus a new ground-state energy. The type of transformation demonstrated here can be applied generally starting with any of the functions Ψ2n(ix) = exp(x 2/2) H2n(ix). 582

d :1 x

dx :1 x

20

2

exp x 2/ 2

2

:1 x

2

This potential would have all of the original harmonic oscillator energy values (2n + 1) from eq 29 and the additional state e = ‒5 having a wavefunction

d dx

d :1 x

dx

10

h2 2m

2 x 2

2x2 1

(38)

u x :1 x 0

d dx

A

2

Energy /

v x x 2 2 1

4

d x2

and

d 2

Now consider a pair of operators A and A+ defined by

A

given as

(37)

u x : x e e1 : x

d T x

dx

v x u x 2

u x : x e : x

d x2

Any such Hamiltonian H and potential u(x) can be “shifted” to define a new Hamiltonian H_ and potential u_(x) = u(x) − e1 such that all the original Ψ(x) for H are also wavefunctions for H_. Specifically, the ground-state wavefunction of the original H and u(x) is also the ground-state wavefunction for the shifted Hamiltonian H_ and u_(x) having energy e = 0:

and ultimately a new potential

d 2

H : x

H : x

2

The Darboux transformation described and applied above in several examples is closely connected to operator based formulations of quantum mechanics. In particular, it is directly associated with factorization (the idea that a Hamiltonian operator can be represented as the product of two operator factors) and the emergent area of supersymmetic quantum mechanics. Consider the general Schrödinger equation in terms of a Hamiltonian operator H and potential u(x):

x 2 exp x 2/ 2 4 x 2 2

2

Connections to Factorization and Supersymmetric Quantum Mechanics

0

u(x) ä x 2

10

v(x) ä ź2 á x2 ź 20

4

2

8 32 x2 á 2x2 á 1 (2x 2 á 1)2 0

2

4

x Figure 3. The harmonic oscillator potential and a related Darboux partner potential v(x). An additional energy state e = –5 has been added to the evenly spaced harmonic oscillator energies.



Applying these operators in the order A+A to a wavefunction Ψ(x) of the original Hamiltonian H gives

d ¯ dx

d :1 x

dx

d dx d : x

2

dx

2

d dx

:1 x

d :1 x

dx

d :1 x

dx :1 x

d : x

dx

:1 x

d 2 :1 x

: x

d :1 x

: x

dx :1 x

tor and the idea of “raising” (creation) and “lowering” (annihilation) operators (14). As the discussion here shows, this notion is much more general, since any Hamiltonian H can be “shifted” to define a Hamiltonian H_. Understanding that the Darboux transformation is equivalent to applying the operator A to a wavefunction and so is related to factorization also helps to show how it is connected to the generation of new potentials and the field known as supersymmetric quantum mechanics. Instead of the factorization A+A = H_, consider the factorization AA+. Applying this to some arbitrary Ψ(x) gives

: x

d x2 :1 x

d :1 x

dx :1 x

2

: x

d :1 x

dx d : x

:1 x

dx

A A : x

d dx

d:1 x

dx d : x

:1 x

dx d :1 x

dx :1 x

d :1 x

dx :1 x

d :1 x

dx

d dx

:1 x

: x

d 2 : x

dx2

d 2 :1( x ) :( x ) d :( x ) dx2 :1 ( x ) dx2 d2 : x

dx 2

d :1 x

dx

d2 : x

dx 2

and d 2 :1 x

d x2

:1 x

can be defined, the operators A and A+ can be defined, and H_ can then be factored as the product A+A. The second term in both A and A+ is exactly the function σ(x) defined in eq 3. This equivalence means that applying the operator A to a wavefunction Ψ(x) is exactly the same as finding the function K x

d : x

dx

d : x

T x : x

dx

the initial step for the Darboux transformation. Chemists usually encounter the idea of factorization of Hamiltonian operators when dealing with the harmonic oscilla-

:1 x

d2 : x

d x2

: x

:1 x

d :1 x

dx

d : x

dx

:1 x

: x

: x

dx2 2 :1 x

u x : x 2

Defining a new potential

2

2

d2 :1 x

d :1 x

: x

dx :1 x

d :1 x

dx

d : x

dx

d :1 x

dx

(39)

:1 x

:1 x

: x

:1 x

: x

dx2 :1 x

d :1 x

dx

In other words, as long as the terms

d dx

d 2 :1 x

u x e1 : x H : x

e e1 : x

d :1 x

dx

2

2

u x u x 2

2

d :1 x

dx

: x

:1 x

d :1 x

dx

2

: x

:1 x

d :1 x

dx

(40)

2

:1 x

then gives a new Hamiltonian H+Ψ(x) = AA+ Ψ(x). From eq 4, since dT x

T 2 x u x e1 dx 2 u x u x 2 T x

dT x

u x e1 2u x 2e1 2 (41) dx d T x

u x e1 2 dx


583


and the definition for the u+(x) for H+ is nearly the same as the definition of the potential v(x) associated with eq 8, except for the constant valued energy e1 term. While reversing the order of the operators A+ and A used in the factorization of H_ generates a new Hamiltonian H+, at this point the wavefunctions associated with the new Hamiltonian have not been found. A clever application of the operator A generates these wavefunctions from the Ψ(x) associated with H_. Apply H+ to a wavefunction defined by A Ψ(x)

H ©« A : x ¸º A A A : x

A A A : x

A H : x

A e e1 : x

(42)

e e1 A : x

in other words, the wavefunctions for H+ are the A Ψ(x), having all the same energies as the wavefunctions for H_, except for the ground state. As noted above, applying A to the Ψ(x) is exactly what the Darboux transformation does! In supersymmetric quantum mechanics, the potentials u_(x) and u+(x) are called “partner” potentials and described in terms of a “superpotential” W(x) defined by d :1 x

dx W x :1 x

(43)

giving dW x

u x W 2 x dx

and

u x W 2 x

dW x

dx

(44)

and operators

A

A

d W x

dx

d W x

dx

The superpotential W(x) is really nothing more than ‒σ(x) as defined by eq 3. This supersymmetric formulation of quantum mechanics is the subject of active current research. It allows for a systematic development of the properties of known solvable potentials and provides a method for determining and characterizing new solvable potentials (e.g., ref 3).

584

Conclusion We have attempted here to give an overview of the Darboux transformation, to show how it can be applied to some quantum mechanical systems familiar to most chemists and to outline the basic connections between this technique and some modern operator based approaches to quantum mechanics. In the interest of clarity, we have tried to be explicit in showing most of the mathematical steps leading to the results, at the expense of length. While involved at times, these steps fundamentally do not require mathematics beyond algebra, single variable calculus, and the use of results available in standard handbooks and texts. Our hope is that through this overview, both students and instructors of quantum chemistry can broaden their understanding of quantum mechanics and be better able to appreciate some of the newer methods being applied. Literature Cited 1. Pronchik, J. N.; Williams, B. W. J. Chem. Educ. 2003, 80, 918–926. 2. Infield, L.; Hull, T. E. Rev. Mod. Phys. 1951, 23, 21–68. 3. Cooper, F.; Khare, A.; Sukhatme, U. Supersymmetry in Quantum Mechanics; World Scientific: Singapore, 2001. 4. Dutt, R.; Khare, A.; Sukhatme, U. P. Am. J. Phys. 1988, 56, 163–168. 5. Bagrov, V. G.; Samsonov, B. F. Pramana 1997, 49, 563–580. 6. Flugge, S. Practical Quantum Mechanics; Springer: Berlin, 1974; pp 89–93. 7. Atkins, P.; Friedman, R. Molecular Quantum Mechanics, 4th ed.; Oxford University Press:Oxford, 2005; pp 55–58. 8. Hochstrasser, U. W. Orthogonal Polynomials. In Handbook of Mathematical Functions; Abramowitz, M., Stegun, I., Eds.; Applied Mathematics Series 55; National Bureau of Standards: Washington, DC, 1964; p 781, eq 22.6.8. 9. Erdelyi, A. Higher Transcendental Functions, Vol. 1; McGraw-Hill: New York, 1953; p 177, eq 3.15(15). 10. Hochstrasser, U. W. Orthogonal Polynomials. In Handbook of Mathematical Functions, Applied Mathematics Series 55; Abramowitz, M., Stegun, I., Eds.; National Bureau of Standards: Washington, DC, 1964; p 774, eq 22.2.3. 11. Oberhettinger, F. Hypergeometric Functions. In Handbook of Mathematical Functions, Applied Mathematics Series 55; Abramowitz, M., Stegun, I., Eds.; National Bureau of Standards: Washington, DC, 1964; p 556, eq 15.1.1. 12. Erdelyi, A. Higher Transcendental Functions, Vol. 1; McGraw-Hill: New York, 1953; p 176, eq 3.15(5). 13. Atkins, P.; Friedman, R. Molecular Quantum Mechanics, 4th ed.; Oxford University Press: Oxford, 2005; pp 60–65. 14. Atkins, P.; Friedman, R. Molecular Quantum Mechanics, 4th ed.; Oxford University Press: Oxford, 2005; pp 521–523.

Supporting JCE Online Material

http://www.jce.divched.org/Journal/Issues/2008/Apr/abs576.html Abstract and keywords Full text (PDF) with links to cited JCE articles


New Potentials for Old: The Darboux Transformation in Quantum

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