Infinite square well, dimensionless Hamiltonian..

In summary, an infinite square well potential is a model used in quantum mechanics to describe a one-dimensional particle confined within a deep potential well. The dimensionless Hamiltonian is a mathematical operator that represents the energy of the particle in relation to its surroundings. The potential well creates a quantized energy spectrum, with the ground state being the lowest energy level. This model is often used to gain understanding of more complex quantum systems.
  • #1
Cocoleia
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Homework Statement
Electron with mass m* moves in a 1D quantum well with infinite barriers at x= -b/2 and
x=+b/2.
Assume mass of the electron to be m*=0.5 m where “m” is free electron mass.
a. Write Schrodinger equation and the Hamiltonian in dimensionless form
b. Solve stationary Schrodinger equation, find allowed energy levels, give explicit expression for 1st,2nd, and 3rd energy levels
d. give wavefunctions for the first three levels – plot them.
e. give 3 energy levels ,in meV, for a=4 nm
Relevant Equations
H phi = -hbar^2 / 2m d^2/dx phi(x) = E phi (x)
I have always seen this problem formulated in a well that goes from 0 to L

I am confused how to use this boundary, as well as unsure of what a dimensionless hamiltonian is.
This is as far as I have gotten
1572389128351.png
 
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  • #2
What would keep you from solving it as if the well was from 0 to ##b## and then, once you had the solution, changing variables ##x' = x-\frac{b}{2}##?
 
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  • #3
Alternatively,put the boundary conditions at -b/2 and b/2 same as you did for [0,L]
 

1. What is an infinite square well potential?

An infinite square well potential is a type of potential energy function commonly used in quantum mechanics to describe a particle confined to a specific region of space. It is characterized by infinitely high walls on either side of the particle's movement, creating a well-like shape.

2. What is a dimensionless Hamiltonian?

A dimensionless Hamiltonian is a mathematical operator used in quantum mechanics to describe the total energy of a system. It is a dimensionless quantity because it does not have any physical units and is often used to simplify calculations.

3. What is the significance of the infinite square well potential?

The infinite square well potential is significant because it serves as a simple and solvable model for understanding more complex quantum mechanical systems. It allows for the study of particle behavior in a confined space and provides insights into energy levels and wave functions.

4. How is the infinite square well potential related to the particle-in-a-box model?

The infinite square well potential is essentially the one-dimensional version of the particle-in-a-box model. In both cases, the particle is confined to a specific region of space with infinitely high walls on either side. However, the particle-in-a-box model has finite walls while the infinite square well potential has infinitely high walls.

5. How does the dimensionless Hamiltonian relate to the Schrödinger equation?

The dimensionless Hamiltonian is a key component of the Schrödinger equation, which is the fundamental equation of quantum mechanics. The Hamiltonian operator, when applied to the wave function of a system, yields the total energy of the system. In the case of the infinite square well potential, the dimensionless Hamiltonian is used to solve for the allowed energy levels and corresponding wave functions.

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