Supersymmetric Quantum Mechanics
of Scattering
Abstract
In the quantum mechanics of collision problems we must consider scattering states of the system. For these states, the wave functions do not remain in Hilbert space, but they are expressible in terms of generalized functions of a Gel’fand triplet. Supersymmetric quantum mechanics for dealing with the scattering states is here proposed.
SUSY (supersymmetry) is a concept which connects between bosons and fermions [1, 2]. SUSY QM (supersymmetric quantum mechanics) is the more elementary concept, because it forms a corner-stone in the theory of SUSY in high-energy physics. An example of a dynamical system in SUSY QM is the SUSY HO (supersymmetric harmonic oscillator) [3, 4]. This example is of a stable system, so the eigenvalue problem is solved in Hilbert space. (For a review of SUSY QM, see for example, reference [5].)
In the present paper a radically different theory for SUSY is put forward, which is concerned with collision problems in SUSY QM. A simple and interesting model of resonance scattering in quantum mechanics is the PPB (parabolic potential barrier) [6, 7, 8, 9, 10, 11]. This model is of importance for general theory, because the eigenvalue problem of the PPB can be solved exactly by a operator method on the same lines as one has used for the HO [12, 13]. We must therefore begin to investigate the SUSY PPB in order to set up the theoretical scheme for dealing with the scattering states of collision problems in SUSY QM.
The parabolic potential barrier
Let us first deal with a PPB [11], a different problem from SUSY problem. The Hamiltonian of the PPB is
(1) |
where and also . The standard states of this PPB have the wave functions, which we may call :
(2) |
These do not belong to a Lebesgue space , but they are generalized functions in of the following Gel’fand triplet [10]:
(3) |
where is a Schwartz space.
Introduce the normal coordinates [6, 8, 9]
(4) |
essentially self-adjoint on . Using the commutation relation , we find
(5) |
It should be noted that the ambiguity of sign in the first of equations (5) is connected with the choice of the arbitrary signs in (4) (cf. reference [11], equations (5) and (7)). We also find that the Hamiltonian (1) is
(6) |
where . Note that the extensions of the normal coordinates operating to a generalized function in have the meaning of operating [11].
Let us now take and operate on them with . Since we have and in the Schrödinger representation, we obtain
(7) |
so that applied to give zero. Then
(8) |
and are generalized eigenstates of belonging to the complex energy eigenvalues . Similarly,
(9) |
Thus the states are generalized eigenstates of belonging to the complex energy eigenvalues .
Take this same Hamiltonian and apply it in the Heisenberg picture. The Heisenberg equations of motion give
(10) |
and the solutions are
(11) |
The time factors of (11) are the same as in the classical theory.
Again, we introduce some essentially self-adjoint operators , to satisfy [11]
(5) |
which are relations of the same form as (5) except for the anticommutators now replacing the commutators there and which therefore contain the ambiguity of sign. Instead of (6) we put
(6) |
where is the fermion number operator
(6) |
The supersymmetric parabolic potential barrier
Let the SUSY Hamiltonian of the SUSY PPB be
(12) |
where is given by (1) or (6) and is given by (6) and (6). Using the values of the commutators and anticommutators given by (5) and (5), we get
and hence the SUSY Hamiltonian (12) is essentially self-adjoint.
Let us consider the essentially self-adjoint operators , defined by
(13) |
Since the time factors in formulas (11) and (11) cancel out in their products of (13), they are constants of the motion. This leads, as will be shown in equations (15), to the result that are the supercharges of the SUSY PPB. We must evaluate the commutators and anticommutators of the supercharges with the normal coordinates , , the SUSY Hamiltonian , and with each other. Using the laws (5) and (5), we obtain
(14) |
and similarly,
(15) |
Again
(16) |
Equations (14) show that make the SUSY transformation which interchanges the operators of “bosonic” and “fermionic”. We have in (16) the SUSY algebra in the SUSY PPB. The first of equations (16), however, does not mean that the SUSY Hamiltonian is positive definite.
We can form the generalized Fock spaces of the SUSY PPB on the same lines as the SUSY HO [3, 4, 5]. We now consider the following states:
(17) |
The right-hand sides here are undetermined to the extent of arbitrary numerical factors. We may consider the states as standard states, since
(18) |
both states have zero energy eigenvalue. Also
(19) |
This shows that the states are supersymmetrical. Thus the standard states , are twofold degenerate. Further, if we shall consider the degree of degeneracy of the fermion sector (caused by the doublets and ), we now have the SUSY-quartet consisting of four kinds of standard states, . It is interesting that such a stable idea as zero energy eigenvalue should appear in the SUSY PPB in this way. These stationary states of the SUSY PPB are analogous to the stationary flows of the 2D PPB [14]. The energy eigenvalues of the other states can be obtained from (17). We have from (9) and (9)
(20) | |||
where | |||
(21) |
Thus the states and are eigenstates of belonging to the same complex energy eigenvalues with , respectively. This result may be verified by (15), since, by applying the supercharges to these states, we can get
Provided that we take account of the twofold degeneracy of the fermion sector, and will in general form quartets for the SUSY PPB.
The superpotential
The above analysis can be extended to the SUSY problem of scattering. We introduce an arbitrary real function which satisfies, as the generalization of (4) and (13),
(22) |
We call the superpotential for the scattering process in SUSY QM, to keep up the analogy with the usual formulation of SUSY QM [3, 4, 5].
The SUSY Hamiltonian for the scattering process is, from the first of equations (16) which are valid also for the general theory,
(23) |
with given by (6). Note that the second term in the brackets in (23), which is the part of referring to the potential energy for the scattering process, appears with a minus sign. One can check that commute with and are constants of the motion.
Let us write the standard states which are supersymmetrical. These states will correspond to wave functions , say, satisfying
(24) |
The first of these equations tells us that the wave functions will be of the form
(25) |
where satisfy (7). With the help of this result the second of equations (24), written in terms of -representatives, becomes
(26) |
Hence we get
(27) |
except for the numerical factors.
Our work on the SUSY PPB in equations (12)–(21) provides an example of a superpotential of resonance scattering. Equation (12) is of the form (23) with for , and it shows that the wave functions (27) agree with (2). It should be noted that represents particles moving outward to the infinity , and represents particles moving inward to the origin [10]. The result for the SUSY PPB is still valid when the superpotential is an odd function of .
On the other hand, when the superpotential is an even function of , the behaviors of the standard states may be changed. Let us see what the above results become in the simple case when (a real constant). Equations (27) for this case read
showing that describes plane waves moving to the -direction, and describes plane waves moving to the -direction. The result for the SUSY free particle is valid whenever the superpotential is an even function of .
The theory that has been set up here is applicable to collision problems in SUSY QM. If we take the usual formulation of SUSY QM [3, 4, 5], we may set up the above-mentioned scheme by taking the superpotential, , in SUSY QM, and replacing it by the method of complex scaling [15].
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