Quantum Particles from Classical Probabilities in Phase Space

Abstract

Quantum particles in a potential are described by classical statistical probabilities. We formulate a basic time evolution law for the probability distribution of classical position and momentum such that all known quantum phenomena follow, including interference or tunneling. The appropriate quantum observables for position and momentum contain a statistical part which reflects the roughness of the probability distribution. “Zwitters” realize a continuous interpolation between quantum and classical particles. Such objects may provide for an effective one-particle description of classical or quantum collective states as droplets of a liquid, macromolecules or a Bose-Einstein condensate. They may also be used for quantitative fundamental tests of quantum mechanics. We show that the ground state for zwitters has no longer a sharp energy. This feature permits to put quantitative experimental bounds on a small parameter for possible deviations from quantum mechanics.

Appendix: Absence of Conserved Energy for Zwitters

In this appendix we argue that for zwitters defined by the particular interpolating Hamiltonian H _γ (94) no conserved energy exists (besides the trivially conserved generator of time translations H _γ itself). This involves the search for an energy operator $\hat{H}$ which commutes with H _γ . We have therefore to explore the commutation relations for different pieces that could be additive building blocks of $\hat{H}$ with H _W and H _L .

With

$$ \begin{array} {@{}rcl} H_L&=&\displaystyle H_{kin}+V_L,\qquad H_{kin} =\frac{1}{2m} \bigl(P^2_Q-\tilde{P}^2_Q\bigr), \\[4mm] V_L&=&\displaystyle V' \biggl(\frac{X_Q+\tilde{X}_Q}{2} \biggr) (X_Q-\tilde{X}_Q), \\[4mm] H_W&=&H_Q-\tilde{H}_Q, \end{array} $$

(140)

one has the commutation relations

$$ \begin{array} {@{}rcl} [H_Q,\tilde{H}_Q]&=&0, \qquad [H_{cl},H_L]=0, \\[4mm] [H_Q,H_L]&=&\displaystyle\frac{1}{2m} \bigl[P^2_Q,V_L\bigr]-\bigl[H_{kin},V(X_Q) \bigr], \\[4mm] [\tilde{H}_Q,H_L]&=&\displaystyle\frac{1}{2m}\bigl[ \tilde{P}^2_Q,V_L\bigr]-\bigl[H_{kin},V( \tilde{X}_Q)\bigr], \end{array} $$

(141)

and

$$ \begin{array} {@{}rcl} [H_Q,H_{cl}]&=& \displaystyle\frac{1}{2m} \biggl[P^2_Q,V \biggl( \frac{X_Q+\tilde{X}_Q}{2} \biggr) \biggr] -\frac{1}{8m} \bigl[ \bigl(P^2_Q+ \tilde{P}^2_Q-2P_Q\tilde{P}_Q \bigr),V(X_Q) \bigr], \\[4mm] [\tilde{H}_Q,H_{cl}]&=&\displaystyle\frac{1}{2m} \biggl[\tilde{P}^2_Q,V \biggl(\frac{X_Q+\tilde{X}_Q}{2} \biggr) \biggr] -\frac{1}{8m} \bigl[\bigl(P^2_Q+\tilde{P}^2_Q-2P_Q\tilde{P}_Q\bigr),V( \tilde{X}_Q) \bigr]. \end{array} $$

(142)

We notice that the last four commutators typically differ from zero such that the existence of a conserved energy is not obvious.

Next we investigate the commutation relations for different possible pieces composing $\hat{H}$. We list

$$ \begin{array} {@{}l} \bigl[P^2_Q,V(X_Q) \bigr]=-2iV'(X_Q)P_Q-V''(X_Q), \\[2mm] \bigl[P_Q\tilde{P}_Q,V(X_Q) \bigr]=-iV'(X_Q)\tilde{P}_Q, \end{array} $$

(143)

and

$$ \begin{array} {@{}l} \displaystyle \biggl[P^2_Q,V \biggl(\frac{X_Q+\tilde{X}_Q}{2} \biggr) \biggr]= -iV' \biggl( \frac{X_Q+\tilde{X}_Q}{2} \biggr)P_Q-\frac{1}{4} V'' \biggl(\frac{X_Q+\tilde{X}_Q}{2} \biggr), \\[4mm] \displaystyle\biggl[P_Q\tilde{P}_Q,V \biggl(\frac{X_Q+\tilde{X}_Q}{2} \biggr) \biggr]=-\frac{i}{2} V' \biggl(\frac{X_Q+\tilde{X}_Q}{2} \biggr) (P_Q+\tilde{P}_Q)-\frac{1}{4} V'' \biggl(\frac{X_Q+\tilde{X}_Q}{2} \biggr), \end{array} $$

(144)

as well as

$$ \begin{array} {@{}lll} \bigl[P^2_Q,V_L \bigr]&=&\displaystyle-2iV' \biggl(\frac{X_Q+\tilde{X}_Q}{2} \biggr)P_Q \\[4mm] &&{}\displaystyle-V'' \biggl(\frac{X_Q+\tilde{X}_Q}{2} \biggr) \bigl(1+i(X_Q-\tilde{X}_Q)P_Q \bigr) - \frac{1}{4}V''' \biggl(\frac{X_Q+\tilde{X}_Q}{2} \biggr) (X_Q-\tilde{X}_Q), \\[4mm] \bigl[\tilde{P}^2_Q,V_L\bigr]&=& \displaystyle2iV' \biggl(\frac{X_Q+\tilde{X}_Q}{2} \biggr)\tilde{P}_Q +V'' \biggl(\frac{X_Q+\tilde{X}_Q}{2} \biggr) \bigl(1-i(X_Q-\tilde{X}_Q)\tilde{P}_Q \bigr) \\[4mm] &&{}\displaystyle-\frac{1}{4} V''' \biggl( \frac{X_Q+\tilde{X}_Q}{2} \biggr) (X_Q-\tilde{X}_Q), \end{array} $$

(145)

and

(146)

For the energy observable $\hat{H}$ we make the ansatz of a linear combination

$$ \hat{H}=\frac{A}{2m}P^2_Q+ \frac{\tilde{A}}{2m}\tilde{P}^2_Q+\frac {B}{m}P_Q \tilde{P}_Q+EV(X_Q) +\tilde{E}V(\tilde{X}_Q)+FV \biggl(\frac{X_Q+\tilde{X}_Q)}{2} \biggr). $$

(147)

This yields the commutation relation with H _γ

(148)

The classical Hamiltonian corresponds to $A=\tilde{A}=1/4$, B=−1/4, $E=\tilde{E}=0$, F=1 and commutes with H _γ for sin² γ=1, while the quantum Hamiltonian obeys A=E=1, $\tilde{A}=B=\tilde{E}=F=0$, and commutes with H _γ for cos² γ=1. For intermediate values of γ and generic unharmonic potentials the relation $[H_{\gamma},\hat{H}]=0$ has no nontrivial solution. Adding further terms to $\hat{H}$ does not change the situation. Generically, no nontrivial conserved energy exists for zwitters of the type defined by Eq. (94).

As an example we may consider the “zwitter energy”

$$ E_\gamma=\cos^2\gamma H_Q+\sin^2\gamma H_{cl}, $$

(149)

which interpolates between the quantum energy for γ=0 and the classical energy for γ=π/2. This corresponds to $A=\cos ^{2}\gamma +\sin^{2}\gamma/4,\tilde{A}=-B=\sin^{2}\gamma/4,~E=\cos^{2}\gamma,\tilde{E}=0,~F=\sin^{2}\gamma$ and we infer the commutator

(150)

Performing a Taylor expansion of V using Eq. (113) we find that in leading order (e=0) only the cubic term contributes to the commutator

$$ [H_\gamma,E_\gamma]=-\frac{id\cos^2\gamma\sin^2\gamma}{16m} (X_Q-\tilde{X}_Q)^2(P_Q+\tilde{P}_Q). $$

(151)

As it should be, the commutator vanishes for γ=0,π/2 and for a harmonic potential.