# I Why not calculate the "trajectory" of a wave function

1. Jul 25, 2017

### zhouhao

The classic limit of Schrodinger equation is hamilton-jacobi eqution.

Wave function's classic limit is $\exp{\frac{i}{\hbar}S(x,t)}$,$S(x,t)$ is the action satisfying hamilton-jaccobi eqution.

However, a particle travels along single trajectory of $S(x,t)$,
Why not make some constrains on wave fucntion to reveal the "single trajectory" from wave function?

2. Jul 25, 2017

### Staff: Mentor

That only works in the classical limit, where you can ignore quantum mechanics.

de-Broglie-Bohm has classical trajectories, but everything is governed by the wave function (or pilot wave), so that doesn't really change anything.

3. Jul 26, 2017

### zhouhao

Thanks.
Classic Mechanic :
1,wave fucntion ($\hbar \rightarrow 0$) is $\psi \rightarrow \exp{\frac{i}{\hbar}S(x,t)}$;
2,we get wave function from Hamilton-Jaccobi eqution with boundary condition,
$\frac{\partial{S}}{\partial{t}}+\frac{1}{2m}{(\frac{\partial{S}}{\partial{x}})}^2+V(x)=0$;
3,$\frac{dx}{dt}=\frac{1}{m}\frac{\partial{S}}{\partial{x}}$ could define a trajectory $x(t)$ for a particle with initial condition $x(0)=a$;

Quantum mechanic:
1,wave fucntion $\psi(x,t)$;
2,we calculate $\psi(x,t)$ from Schrodinger eqution, $-\frac{{\hbar}^2}{2m}\frac{{\partial}^2{\psi}}{\partial{t}^2}+V(x)=i\hbar\frac{\partial{\psi}}{\partial{t}}$;
My textbook stop at second step , just calculate wave function and ignore the third step in classic mechanic which make me confused.
Maybe third step in QM, could be like this -----
-----3,$(\frac{dx}{dt})_n=\frac{(\frac{\hat{p}}{m})^n\psi}{(\frac{\hat{p}}{m})^{(n-1)}\psi}$,$n \ge 1$ is an integer.$\hat{p}=-i\hbar\frac{\partial}{\partial{x}}$

When $\hbar \rightarrow 0$ , $(\frac{dx}{dt})_n \rightarrow \frac{1}{m}\frac{\partial{S}}{\partial{x}}$
This means we define many trajectory for a particle.

4. Jul 27, 2017

### Staff: Mentor

You can define a lot of things, that doesn't mean they have to have useful properties. I didn't study your definition in detail, but if in doubt, it won't lead to a continuous trajectory, or even to ill-defined expressions.

5. Jul 27, 2017

### Demystifier

https://en.wikipedia.org/wiki/De_Broglie–Bohm_theory

6. Jul 30, 2017

### zhouhao

Thanks.I think Bohmian mechanics is helpful to me.Could help me with another question below?
$\psi$ is a solution of Schrodinger eqution.
When $\hbar \rightarrow 0$,${\psi}(x,t) \rightarrow {\rho}(x,t)e^{\frac{i}{\hbar}S(x,t)}$
Define $\frac{dq}{dt}=\frac{1}{m}\frac{\partial{S}}{\partial{x}}$,
and $\delta{(x-q(t))}$ means choose trajectories beside $q(t)$,not the real Dirac function.
The ${\psi}_q \rightarrow \delta{(x-q(t))}e^{\frac{i}{\hbar}S(x,t)}$ is a solution.
The linear combination ${\sum\limits_q}c_q{\psi}_q$ is also a solution.
Too many solutions.
If we calculates wave fucntion of electron moving around nuclear or the one of electron in the two-slit diffraction experiment,how to get the initial boundary couditon?Is there any example to calculate this kind of thing?

7. Aug 2, 2017

### Demystifier

There is no simple recipe hot to find the right solutions (right initial conditions). You must study a whole textbook to learn something about it.