Minimum Uncertainty in Electron Position: Rectangular Wave in k-space

In summary, the electron wave packet has a rectangular profile in the k-space, with a minimal uncertainty in position of $\frac{1}{2a}$ and a wavefunction given by $\psi(x,t) = A \sin{(2\pi(\frac{k}{2\pi}x-vt))}$.
  • #1
The Head
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Homework Statement


In the case of an electron wave packet, the function A(k) has a rectangular shape, i.e. it is equal to A0 if k0-a<k<k0+a, and zero everywhere else. (a) Find the minimal uncertainty of electron position. (b) Find the electron wavefunction.


Homework Equations


ΔxΔp=h/4pi (for min uncertainty)
ΔxΔκ=1/4pi
ψ=Asin2pi(κx-vt)

The Attempt at a Solution


a) κ=1/λ
ΔxΔκ=ΔxΔ(1/λ)=1/4pi
Δx=1/(4pi(Δ(1/λ))= 1/(4pi(Δ(k/2pi))
Because k0+a -(k0-a)=2a, the span of k is 2a?
Δx=2pi/(4pi*2a)=1/2a
I am not sure if I am doing this correctly. My lecture notes talk about k and λ space, making no mention of κ, but my book uses equations with kappa. It seems like κ=k/2pi, but maybe I have that wrong.

b)ψ=Asin2pi(κx-vt) Not sure where to go from here

I appreciate any help I can get!
 
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  • #2
For part a), you are correct in using the uncertainty principle, and it is true that $\Delta k = 2a$, so $\Delta x = \frac{1}{2a}$. For part b), remember that the wavefunction is a function of space and time. You know the functional form already (the sinusoidal form with a phase factor). The amplitude A is determined by the function A(k) which you are given, so you just need to plug in the right values for k, x, and t.
 

Related to Minimum Uncertainty in Electron Position: Rectangular Wave in k-space

1. What is the minimum uncertainty in electron position in a rectangular wave in k-space?

The minimum uncertainty in electron position in a rectangular wave in k-space is given by the Heisenberg uncertainty principle, which states that the product of the uncertainty in position and momentum of a particle must be greater than or equal to the reduced Planck's constant divided by 2π.

2. How is the minimum uncertainty in electron position related to the size of the wave in k-space?

The minimum uncertainty in electron position is inversely proportional to the size of the wave in k-space. This means that the smaller the wave, the larger the uncertainty in position, and vice versa.

3. What is the significance of the minimum uncertainty in electron position in a rectangular wave in k-space?

The minimum uncertainty in electron position in a rectangular wave in k-space is significant because it sets a limit on the precision with which we can measure the position and momentum of a particle. It also highlights the inherent uncertainty and randomness in the behavior of subatomic particles.

4. Can the minimum uncertainty in electron position be reduced or eliminated?

No, the minimum uncertainty in electron position cannot be reduced or eliminated. It is a fundamental property of quantum mechanics and is not due to limitations in our measurement techniques.

5. How does the minimum uncertainty in electron position affect our understanding of the behavior of electrons?

The minimum uncertainty in electron position challenges our traditional understanding of particles as having well-defined positions and momenta. It suggests that at the subatomic level, particles behave more like waves and have inherent uncertainty in their properties. This has significant implications for our understanding of the nature of reality and the fundamental laws of the universe.

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