Effect of momentum distribution on probability density

[WWCY]

Hi all, I have the following query:

I understand that the "make-up" of momentum probability density ##|\tilde{\Psi}|^2## has an effect on the motion of the spatial probability density ##|\Psi|^2##. For example, a Gaussian ##|\tilde{\Psi}|^2## centred far to the right will cause ##|\Psi|^2## to travel right over time.

However, is there any way to pinpoint the effects that a certain range of ##|\tilde{\Psi}|^2## has on ##|\Psi|^2##? For example, could one find out what effects momenta ##p ∈ [p_1,p_2]## have on the motion of ##|\Psi|^2##?

Thanks in advance!

 

[DrClaude]

In momentum space, you basically have a superposition of plane waves, so each ##\vec{p}## component represents a plane wave moving with momentum ##\vec{p}##.

You can also take a Fourier transform of the ##[p_1,p_2]## range at two different times to see how it affects the motion of the wave packet.

 

[blue_leaf77]

Imagine at t=0 your momentum space wavefunction are real, this means all of the plane waves are in phase at the start. Then as time passes, each plane wave accumulates phase at different rate since they move with different velocities. Mathematically at arbitrary time ##t##, you have ##\tilde\psi(p,t) = \exp(-ip^2t/(2m)) \tilde\psi(p,0)##.
In position space it is intuitive to think that at any given time ##t>0## the rear portion (in x axis, the left part) of ##\psi(x,t)## must be dominated by the lower momentum part of ##\tilde\psi(p,t)## because this portion moves slower than the rest of the plane waves. On the other hand, the front part of ##\psi(x,t)## must be dominated by the higher momentum part of ##\tilde\psi(p,t)##.

However, is there any way to pinpoint the effects that a certain range of |~Ψ|2|\tilde{\Psi}|^2 has on |Ψ|2|\Psi|^2? For example, could one find out what effects momenta p∈[p1,p2]p ∈ [p_1,p_2] have on the motion of |Ψ|2|\Psi|^2?

I have not tried the following but I think you can follow it since it's interesting to see. At any given time, you can try to calculate the average position of the wavepacket portion contributed by certain momentum range ##[p_a,p_b]##. In other words calculate the following expectation value
$$
x(p_0,t)=\int dp \ \tilde\psi^*(p,p_0,t) i\hbar\partial_p \tilde\psi(p,p_0,t)
$$
with
$$
\tilde\psi(p,p_0,t) = F(p,p_0)\tilde\psi(p,t) = F(p,p_0) \exp(-ip^2t/(2m)) \tilde\psi(p,0)
$$
where ##F(p,p_0)## is the filter/window function centered at ##p=p_0## whose width determines the momentum range of interest.
##x(p_0,t)## above is a function of two variables, I imagine for a fixed ##t## you will get a monotonically increasing function between ##x## and ##p_0## because smaller momenta (slower plane waves) populate smaller ##x##. But you can plot ##x(p_0,t)## as a 2D colormap plot with x and y-axis the variables and ##x## represented by color.

You can also take a Fourier transform of the [p1,p2][p_1,p_2] range at two different times to see how it affects the motion of the wave packet.

In that case and in relation to the method I suggested above, the window function is equal to a box function. The Fourier transform of the resulting windowed momentum wavefunction, i.e. the filtered position space wavefunction, will be a convolution between the a sinc function and the unfiltered position space wavefunction. Such function contains oscillation which attenuates very slowly and therefore can be affected by the range used in the actual computation. In my experience, such windowing function can be well represented by a Gaussian function centered at ##p_0## and standard deviation being approximately equal to ##p_a-p_b##.

 

[WWCY]

Thanks very much to you both for your insights!

 

[vanhees71]

Hi all, I have the following query:

I understand that the "make-up" of momentum probability density ##|\tilde{\Psi}|^2## has an effect on the motion of the spatial probability density ##|\Psi|^2##. For example, a Gaussian ##|\tilde{\Psi}|^2## centred far to the right will cause ##|\Psi|^2## to travel right over time.

However, is there any way to pinpoint the effects that a certain range of ##|\tilde{\Psi}|^2## has on ##|\Psi|^2##? For example, could one find out what effects momenta ##p ∈ [p_1,p_2]## have on the motion of ##|\Psi|^2##?

Thanks in advance!

Without giving the Hamiltonian, I can't tell you anything about how the wave packet moves (neither in momentum nor in position representation). For a free particle, momentum is conserved, and thus the momentum-space wave function doesn't change up to a phase factor and thus the probability distribution is time independent, as it must be.