Infinite Well Problem - Time Independent Schrodinger's Equation

In summary, the conversation discussed Schrodinger's Wave Equation, specifically the 1 dimensional time independent form, in a Semiconductor class. The topic of the conversation was the infinite potential well model, which is divided into three regions: V(x) = -inf on the left, V(x) = 0 in the middle, and V(x) = inf on the right. The second region was solved using the equation d2\Psi(x)/dx2 + \frac{2m}{\hbar}*(E-V(x))*\Psi(x) = 0, with the solution being \Psi(x) = A1cos(kx) + A2sin(kx). The conversation also mentioned that basic knowledge of solving differential
  • #1
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I'm currently taking a Semiconductor class and we're talking about Schrodinger's Wave Equation, specifically the 1 dimensional time independent form.

We were looking at the infinite potential well model:

220px-Infinite_potential_well.svg.png


And we divided the graph into 3 different regions: first being the left (or negative) V(x)= -inf, the second being V(x)=0, and the third being V(x)= inf.

We solved the second region first using the equation:

d2[itex]\Psi[/itex](x)/dx2 + [itex]\frac{2m}{\hbar}[/itex]*(E-V(x))*[itex]\Psi[/itex](x) = 0

Well my professor said this math should be something we could do easily, so pardon me if I seem a bit ignorant, but I really can't recall an effective way to tackle this problem. He gave us the solution, which is:

[itex]\Psi[/itex](x) = A1cos(kx) + A2sin(kx)

I should have included all the information needed, but if not please ask!

Thanks!
 
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  • #3

1. What is the Infinite Well Problem in the context of the Time Independent Schrodinger's Equation?

The Infinite Well Problem is a theoretical concept in quantum mechanics that involves a particle confined within an infinitely deep potential well. This problem is often used to demonstrate the properties of wave functions and energy levels in quantum systems.

2. What is the mathematical expression for the Time Independent Schrodinger's Equation?

The Time Independent Schrodinger's Equation is a partial differential equation that describes the evolution of a quantum system over time. Its mathematical expression is:
HΨ = EΨ
where H is the Hamiltonian operator, Ψ is the wave function, and E is the energy of the system.

3. How is the Infinite Well Problem solved using the Time Independent Schrodinger's Equation?

The Infinite Well Problem is solved by applying the boundary conditions that the wave function must be zero at the edges of the well. This leads to a discrete set of energy levels that the particle can occupy within the well. The solutions to the Schrodinger's Equation are known as eigenfunctions, and the corresponding energies are known as eigenvalues.

4. What are the implications of solving the Infinite Well Problem using the Time Independent Schrodinger's Equation?

Solving the Infinite Well Problem using the Time Independent Schrodinger's Equation allows us to better understand the behavior of quantum particles in confined systems. It also provides insights into the quantization of energy levels and the probability distribution of finding a particle within the well.

5. Can the Time Independent Schrodinger's Equation be applied to other systems besides the Infinite Well Problem?

Yes, the Time Independent Schrodinger's Equation is a fundamental equation in quantum mechanics and can be applied to a wide range of systems, including atoms, molecules, and solids. It is a powerful tool for understanding the behavior of particles at the quantum level and has many practical applications in science and technology.

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