# Bohm Trajectories for a Type of Derivative Nonlinear Schrödinger Equation

Bohm Trajectories for a Type of Derivative Nonlinear Schrödinger Equation

The time-evolution of the quantum standard wavefunction is determined by the Schrödinger equation and guidance equation. The guidance equation states that the velocity field for the configuration is given by the quantum current divided by the density . The guidance equation is derived from the continuity equation, which is a special form of a conservation law.

v

j

ϱ

The velocity field as given for the Schrödinger equation cannot be applied in general. For some nonlinear Schrödinger equations, an acceptable velocity field must be derived from the continuity equation with a modified current. The new velocity field is then postulated to equal the current divided by the density.

Such an example is presented here. In this Demonstration the current is derived from a type of Derivative Nonlinear Schrödinger Equation (DNLSE). The DNLSE is defined by , where is a complex function that depends on , and and is the complex conjugate.

iu+u+iuu+u=0

∂

t

∂

{x,2}

∂

x

*

u

1

2

2

(u)

*

u

u

x

t

*

u

The modified current is obtained from the continuity equation ϱ+j=0, where the current with the density . For the DNLSE case, the velocity must be changed to to satisfy the continuity equation. The guidance equation (here with =0.5 and =1) was developed by de Broglie in the 1920s for the Schrödinger equation and was first introduced at the famous Solvay conference in 1927. The resulting dynamics from the guidance equation were called "pilot wave theory" by de Broglie. The pilot wave theory was rediscovered by David Bohm in 1952.

∂

t

∂

x

j=-i(u-u)+=ϱv

*

u

∂

x

∂

x

*

u

1

2

2

ϱ

ϱ=u

*

u

v

v=-iu-+u

∂

x

u

∂

x

*

u

*

u

1

2

*

u

v=-iu-

∂

x

u

∂

x

*

u

*

u

m

h