Infinite Well Problem - Time Independent Schrodinger's Equation

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SUMMARY

The discussion focuses on the time-independent Schrödinger's Wave Equation as applied to the infinite potential well model in quantum mechanics. The equation used is d²Ψ(x)/dx² + (2m/ħ)(E-V(x))Ψ(x) = 0, with the solution expressed as Ψ(x) = A1cos(kx) + A2sin(kx). The graph is divided into three regions: V(x) = -∞, V(x) = 0, and V(x) = ∞. A basic understanding of differential equations is essential for tackling this problem effectively.

PREREQUISITES
  • Understanding of Schrödinger's Wave Equation
  • Knowledge of quantum mechanics principles
  • Familiarity with differential equations
  • Basic concepts of potential wells in physics
NEXT STEPS
  • Study the derivation of the time-independent Schrödinger equation
  • Learn about boundary conditions in quantum mechanics
  • Explore the implications of the infinite potential well model
  • Investigate solutions to other forms of potential wells
USEFUL FOR

Students in physics or engineering, particularly those studying quantum mechanics, and anyone interested in understanding the mathematical foundations of the Schrödinger equation.

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I'm currently taking a Semiconductor class and we're talking about Schrödinger'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\Psi(x)/dx2 + \frac{2m}{\hbar}*(E-V(x))*\Psi(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:

\Psi(x) = A1cos(kx) + A2sin(kx)

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

Thanks!
 
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