Probability current density

In summary, if a particle's quantum state is described by a single eigenfunction, the probability current density inside the well will vanish due to the time-independence of the wavefunction.
  • #1
stunner5000pt
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Consider the infinite square well potentail over the range 0<x<a with energy eigenfunctions [itex]Psi _{n} (x,t)[/itex]

Show that if the quantum state of a particle is descirbed by a single eigenfuncation i.e. the particle has a sharply defined eneryg, then the probability current density inside the well vanishes

ok Delta E = 0

[tex] \Psi (x,t) = \psi(x) \exp(\frac{-iEt}{m} [/tex]

[tex] j= \frac{\hbar}{2im} \left( \Psi^* \frac{\partial \Psi}{\partial x} - \frac{\partial \Psi^*}{\partial x} \Psi \right) [/tex]

ok let's say that [tex] P(x,t) = | \Psi(x,t)|^2 = \psi^*(x) \psi(x)[/tex]

we also know that \frac{\partial}{\partial t} (\psi^* \psi)= -\frac{\hbar}{2im} \left( \Psi^* \frac{\partial \Psi}{\partial x} - \frac{\partial \Psi^*}{\partial x} \Psi \right) [/tex]

and hence [tex] \frac{\partial}{\partial t} P(x,t) = \frac{\partial}{\partial t}( \psi^*(x) \psi(x) )= 0 = -\frac{\partial}{\partial t} j(x,t) = 0 [/tex]

so the position derivative of probability current wrt position is zero

but does that show that it vanishes??

thank you for your input
 
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  • #2
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Yes, that does show that the probability current density inside the well vanishes. This is because if the quantum state of a particle is described by a single eigenfunction, it means that the particle has a sharply defined energy. This also means that the wavefunction is time-independent, as shown in the equation \Psi (x,t) = \psi(x) \exp(\frac{-iEt}{\hbar}). Therefore, the time derivative of the probability density is equal to zero, and as a result, the probability current density inside the well also vanishes. This indicates that the particle is not moving and is confined within the well, in accordance with the infinite square well potential.
 

FAQ: Probability current density

What is probability current density?

Probability current density is a concept in quantum mechanics that describes the flow of probability in a system. It is a vector field that represents the rate at which the probability of finding a particle at a given point changes over time.

How is probability current density calculated?

Probability current density is calculated using the continuity equation, which states that the change in probability density over time is equal to the negative divergence of the probability current density. This can be expressed mathematically as J = -dρ/dt, where J is the probability current density and ρ is the probability density.

What is the significance of probability current density?

Probability current density is a fundamental concept in quantum mechanics and is used to describe the behavior of particles at the subatomic level. It allows us to understand the flow of probability in a system and make predictions about the behavior of particles.

How does probability current density relate to quantum tunneling?

Probability current density plays a key role in the phenomenon of quantum tunneling, which is when a particle is able to pass through a potential barrier that would be impossible according to classical mechanics. This is because the probability current density allows for the particle to have a non-zero chance of existing on the other side of the barrier.

Can probability current density be observed experimentally?

Yes, probability current density can be observed experimentally using techniques such as scanning tunneling microscopy and electron diffraction. These methods allow scientists to indirectly measure the flow of probability in a system and confirm the predictions made by quantum mechanics.

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