Finite Well and Schrodinger's Equation

In summary, the conversation discusses the calculation of the probability for finding an electron in the left half of a finite well based on its energy and wavefunction. The conversation includes the use of Schrodinger's equation and the calculation of k, which is found to be different by a factor of \sqrt{2 \pi} between the two individuals involved in the conversation. The equation for k is clarified and the reason for the difference in values is addressed.
  • #1
yayz0rs
1
0

Homework Statement


An electron is trapped in a finite well of width 0.5 nm and depth of 50 eV. The wavefunction is symmetric about the center of the well (x = 0.25 nm). If the electron has energy 29.66 eV and ψ(0) = 1.42 (nm)-1/2, then what is the probability for finding the particle in the left half of the well (0 < x < 0.25 nm)?


Homework Equations


Schrodinger's Equation in the classically forbidden zone: C1ekx + C2e-kx

The Attempt at a Solution


My idea was that because the probability density from negative infinity to .25 nm (L/2) would be .5, I could simply find p(<0) by integrating Schrodinger's equation and subtracting that from .5. However I cannot find K.

I know the equation for K is [tex]\sqrt{2m/(h/2\Pi)^2(U-E)}[/tex] and it seems like I've been given those 4 variables. However my homework will not accept my answer for K. For reference I've been getting k = 9.25 nm^-1.
 
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  • #2
Hello yayz0rs,

Welcome to Physics Forums.

I'm getting a different answer for k.

My solution to Shrodinger's equation in that region matches yours (at least involving k). There's nothing wrong with your equation for k, as far as I can tell. But still I'm getting a different value than yours when calculating the number.

For your reference, your value for k is different from mine by close to a factor of [tex] \sqrt{2 \pi} [/tex]. Are you sure you are squaring the [tex] 2\pi [/tex] along with the h? Are you making sure to use the correct value for h (as opposed to [tex] \hbar [/tex] which as the [tex] 2 \pi [/tex] already factored in)?
 
  • #3
By the way, your equation, as it is written, could use a little clarity by adding parenthesis or something appropriately. In my last post, above, I assume your equation really means

[tex] k = \sqrt{\frac{2m}{\hbar^2}(U-E)} [/tex]

where

[tex] \hbar = \frac{h}{2\pi} [/tex]

in which case, we both got the same equation for k.
 

1. What is a finite well in quantum mechanics?

A finite well is a potential energy barrier that is bounded by walls on either side. This represents a region in space where a particle is confined and can only exist within a certain range of energies.

2. How is Schrodinger's equation used to describe a finite well?

Schrodinger's equation is a mathematical equation used to describe the behavior of particles in quantum mechanics. In the case of a finite well, it is used to calculate the energy levels and wavefunctions of a particle confined within the well.

3. What is the difference between an infinite and finite well?

Infinite wells have infinitely high potential energy barriers, meaning that the particle is completely confined and cannot escape. Finite wells have a finite potential barrier, allowing the particle to have a small probability of tunneling through the barrier.

4. How does the depth of a finite well affect the energy levels of a particle?

The depth of the finite well directly affects the energy levels of a particle. A deeper well results in more closely spaced energy levels, while a shallower well leads to more widely spaced energy levels.

5. Can Schrodinger's equation be applied to other systems besides finite wells?

Yes, Schrodinger's equation can be applied to any system that exhibits quantum behavior, such as atoms, molecules, and solid-state materials. It is a fundamental equation in quantum mechanics and is widely used in many areas of physics and chemistry.

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