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I have heard the term quantum leap to describe the (instantaneous?) transition from a higher energy orbital to a lower energy orbital. Yet, I understand that this transition time has now been quantified in certain experiments.

So, it is not a leap? Is the electron wavefunction defined during this transition?

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Comment to the question(v1): To focus the question, it would be great if you could include a reference, either to where you learned this, or to the quantifying experiments. –  Qmechanic Mar 3 '12 at 21:49

2 Answers 2

up vote -1 down vote accepted

So, it is not a leap? Is the electron wavefunction defined during this transition?

Let $\vert \psi_+\rangle$ and $\vert\psi_-\rangle$ the wavefunctions of the electron before, and after the quantum transfer. You therefore have a superposition of the wavefunctions

$\vert \Psi \rangle = a(t) \vert \psi_+\rangle + b(t)\vert\psi_-\rangle$ and we know that the norm of that superposition must be 1. We also can assume

$\vert \Psi_{t=0} \rangle = \vert \psi_+ \rangle $

and

$\vert \Psi_{t=T} \rangle = \vert \psi_- \rangle $

Due to the constraint of $\Vert \vert \Psi \rangle \Vert = 1$ we therefore know that $a(t)^2 + b(t)^2 = 1 \Rightarrow b(t)^2 = 1 - a(t)^2$

In the case of "leap" the function a, b are step functions. But instead of step any function goes for which $a(t)^2 + b(t)^2 = 1$. That could be sin and cos for example, but there a few others as well.

There you have it: The wave function of the electron in transistion state is the superposition of the wavefunctions before and after the transition, weighted by a transfer function. To determine the exact nature of the transfer function one has to dive into QED.

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Is "You therefore have a superposition of the wavefunctions" a priori justified? Otherwise you basically give the answer at that point. On the other hand, I don't really know the dinfinition of the "quantum leap" as I expected the answer to involve an observation process. –  NikolajK Feb 28 '12 at 18:31
    
@NickKidman: In terms of QED the wavefunction of a particle is the superposition of all possible states it can be at any given time, and what changes is the amplitudes of each of those elementary wavefunctions. Also try to understand a "quantum leap" as a tunnel process, specifically not as a tunnel process between orbital states, but tunneling into a free electron, borrowing energy from a virtual photon, then tunneling back into a lower energy state, giving back energy to the virtual photon and emitting the difference as a real photon. –  datenwolf Feb 28 '12 at 18:38
    
Also the abstraction of a particle being in only a pure state is a mathematical idealization. In practice every real wavefunction is a superposition of. All those corrections found in QED and QCD are nothing else, than the contributions of additional superimposing wavefunctions to the ideal, pure state of the simple model. –  datenwolf Feb 28 '12 at 18:41
    
In QED, is it the interference of Phi+ and Phi- during the transition in a sense that produces the outgoing photon packet? –  sonardude Feb 28 '12 at 18:45
    
@sonardude: Wow, good question. I honestly never considered this problem yet. I've also to admit, I'm "only" a experimentalist. But next time I have a chance to talk to somebody of the QED theory group I'll tap his or her brain about this. But I wouldn't be surprised, that if you evaluated the interference of the wavefunctions you'd see some oscillation in the probability flux that would act like a emitter for a photon. –  datenwolf Feb 28 '12 at 18:51

If two otherwise independent systems interact for a short time then they have a definite state before and after the interaction but not during it. During the interaction, only the joint system consisting of both have a well-defined state and evolution (in the tensor product of the two state spaces).

This joint evolution begins with psi_1 tensor psi_2 and ends with a superposition of many of these, of which one will be realized with the probability computed by Born's rule.

Before the interaction is completed and Nature made the choice of which result to produce, there is only the superposition. None of the possibilities participating in the superposition is definite before completion of the interaction and the subsequence collapse (due to the interaction with theenvironment) - so assigning wave functions during this time is completely arbitrary and hence spurious.

In particular this is the case for a quantum leap. The system has well-defined states before and after the jump, but during the jump only the state of a bigger system incorporationg at least some degrees of freedom from the electromagnetic field is meaningful.

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