What is the uncertainty of an electron's momentum in an infinite potential well?

[Feynmanfan]

At t=0 an electron in an infinite potential well has a wave function corresponding to the lowest level of energy. The wave function is equal to the eigenfunction of the Hamiltonian where n=1.

I am asked to calculate the uncertainty of the electron's momentum. I don't really know where to start from.

Thanks for your advice

 

[dextercioby]

How about the definition of the uncertainty...?

Daniel.

 

[thepatientmental]

.

To calculate the uncertainty of the electron's momentum, you can use the uncertainty principle, which states that the product of the uncertainties in position and momentum must be greater than or equal to Planck's constant divided by 2π. In this case, since the electron is in the lowest energy level (n=1) of an infinite potential well, the wave function can be described by a sine function with a wavelength equal to the width of the well. This means that the position of the electron is well defined within the well, but its momentum is uncertain.

To calculate the uncertainty in momentum, you can use the formula Δp = h/λ, where Δp is the uncertainty in momentum, h is Planck's constant, and λ is the wavelength of the electron's wave function. In this case, since the electron's wave function corresponds to the lowest energy level (n=1), the wavelength can be determined from the de Broglie relation, which states that the wavelength of a particle is equal to h/p, where p is the momentum of the particle. Since the electron is in the lowest energy level, its momentum can be determined from the energy eigenvalue of the system, which is given by E = p^2/2m, where m is the mass of the electron. Putting these equations together, we get:

Δp = h/(h/p) = p

Therefore, the uncertainty in momentum is equal to the momentum itself. This means that the uncertainty in momentum is not a fixed value, but rather depends on the value of the electron's momentum. In general, the uncertainty in momentum will be smaller for particles with larger momentum and vice versa.

In summary, the uncertainty principle tells us that the product of the uncertainties in position and momentum is always greater than or equal to Planck's constant divided by 2π. In the case of an electron in an infinite potential well in the lowest energy level, the uncertainty in momentum is equal to the momentum itself, which can be determined from the energy eigenvalue of the system. I hope this helps guide you in your calculations.