system
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human
Does the change in an electron’s wave function and probability density function happen instantaneously when it changes energy state?

Edit: Specifically for the hydrogen atom for simplicity’s sake. 

Wondering if this change happens at the speed of light or instantaneously and if instantaneously how does it not violate causality. Does it work like wave function collapse?
gpt
No, it doesn’t happen instantaneously. For example if you have a particle in an excited state with one lower-energy state (the ground state), the time-dependence of the wavefunction is something like exp[iEt/~~h~~], where E is a “complex-valued energy”

E = E*_0_* - iΓ/2.

E*_0_* is the actual energy of the excited state, and Γ is the energy width. 

If you calculate a probability density for a wavefunction of this form, you get something proportional to

exp[-Γt/~~h~~].

Notice that this is not an imaginary exponential, which is an oscillating function. This is a **real** exponential function, and it decays with time. This is the exact same exponential decay that you see in radioactivity.

So the probability density for finding the particle in the excited state decays exponentially in time; it doesn’t instantaneously change.