John von Neumann carefully distinguished between the two types of quantum evolution processes (Type 1 "quantum jumps" = "wave function collapse"; Type 2 SWE). Now, the probability amplitude, for a "quantum jump", is equal to the overlap integral ([math]< \Phi | \Psi >[/math]), between the quantum particle's current state ([math]\Psi[/math]), and the available end state ([math]\Phi[/math]). (The probability is the squared modulus of the probability amplitude.)

 

QUESTION: Could you view "business as usual", Type 2, SWE-compliant, evolution, as a series of "self-collapses" of the wave function ([math]\Psi(t) \rightarrow \Psi(t+\delta t)[/math]) ? If:

 

[math]\Psi(t + \delta t) \approx \Psi(t) + \frac{\partial \Psi(t)}{\partial t} \delta t[/math]

 

[math] = \Psi(t) - \frac{i}{\hbar}\hat{E} \Psi(t) \delta t[/math]

and if the wave function was normalized ([math]< \Psi | \Psi > = 1[/math]), then wouldn't the probability, for a "self jump" process be:

 

[math]< \Psi(t+\delta t) | \Psi(t) > \; \approx \; 1 - \frac{i <E>}{\hbar} \delta t[/math]

 

[math] \equiv 1 - i <\omega> \delta t[/math]

?

 

In-so-far as such a "self-collapse" probability is [math]< 1[/math], what would happen, on those rare occasions, when the SWE "mis-fired" ([math]\propto - i <\omega> \delta t[/math]) ? Presumably, [math]\delta t \approx t_{Planck}[/math] ?? Could such "mis-fires", of the SWE, explain "spontaneous collapse" theories of quantum jumps ???

 

 

 

Comparison of "self-collapse" to "standard-collapse" probabilities (??)

 

If the probability amplitude, of an "in-jump" process, is [math]P \left( \Psi(t) \rightarrow \Psi(t+\delta t) \right) \; \approx \; 1 - i <\omega> \delta t[/math], would the probability of an "out-jump" process grow as:

 

[math]P \left( \Psi \rightarrow \Phi \right) = < \Phi | \Psi >[/math]

 

[math] \approx < \Phi | \Psi(t) > - \frac{i < \Phi | \hat{E} | \Psi(t) >}{\hbar} \delta t[/math]

 

[math]= a \left( 1 + \frac{i E_{\Phi}}{\hbar} \delta t \right)[/math]

 

[math] \equiv a \left( 1 + i \omega_{\Phi} \delta t \right)[/math]

where we have defined the transition probability amplitude [math]a \equiv < \Phi | \Psi >[/math], and then "acted to the left", with the Energy Operator... ?? Does this mean, accumulating said contributions over many time steps, that all transition probability amplitudes grow exponentially, as [math]e^{i \omega_{\Phi} t}[/math] ???

Edited September 14, 201015 yr by Widdekind

John von Neumann carefully distinguished between the two types of quantum evolution processes (Type 1 "quantum jumps" = "wave function collapse"; Type 2 SWE). Now, the probability amplitude, for a "quantum jump", is equal to the overlap integral ([math]< \Phi | \Psi >[/math]), between the quantum particle's current state ([math]\Psi[/math]), and the available end state ([math]\Phi[/math]). (The probability is the squared modulus of the probability amplitude.)

 

Projection [math]< \Phi | \Psi >[/math] or the amplitude measurement is not an evolution or a collapse.

Edited September 14, 201015 yr by Bob_for_short