Simple Finite Square Well Problem help *Ignore, made stupid mistake*

In summary, the conversation is about solving a second-order differential equation with the assumption that the solution has the form ##e^{\gamma x}##. After substituting the form into the equation, it is found that the solution is either ##\gamma = 0## or ##k^2##. However, there is confusion about how to convert the solution into the form ##c_1 \sin{kx} + c_2 \cos{kx}##, as the real part of the solution does not have an imaginary number. It is later realized that the solution was incorrect due to a mistake in using ##+k^2 \psi## instead of ##\psi'' + k^2 \psi##.
  • #1
Irishdoug
102
16
Homework Statement
Self studying Quantum Mechanics. I've a finite square well. Inside the well the V(x) = 0. Thus the problem becomes that of the free particle. I'm aware the solution to the SE is Asinkx + Bcoskx however I can't figure out how.
Relevant Equations
##\psi ##'' = ##-k^2 \psi ## were k = ##(2mE)^{0.5}## / ##\hbar##
I've tried to carry out the solution to this as a normal 2nd order Differential Equation
##\psi ##'' - ##-k^2 \psi ## = 0
Assume solution has form ##e^{\gamma x}##
sub this in form ##\psi## and get
##\gamma ^2## ##e^{\gamma x} ## + ##k^2 e^{\gamma x}## = 0
Solution is ##\gamma## = 0 or ##k^2##
Now have two real roots that are not equal thus have
c1##e^{\gamma_1 x}## + c2##e^{\gamma_2 x}##
I presume this is wrong, as I cannot figure out how to turn it into c1sinkx +c2coskx.
I'm aware of Eulers equation but I've no imaginary number and just using the real part means I've no sin term.
 
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  • #2
Irishdoug said:
##\gamma ^2## ##e^{\gamma x} ## + ##k^2 e^{\gamma x}## = 0
Solution is ##\gamma## = 0 or ##k^2##
Check your solution to this equation.
 
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  • #3
When I did it I used ##\psi##'' +##k^2 \psi##' instead of simply ##+k^2 \psi## hence my wrong answer. Only realized when I'd written all that out the mistake I'd made. Tis the problem with studying after a full days work!

Cheers for the reply though.
 
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Related to Simple Finite Square Well Problem help *Ignore, made stupid mistake*

1. What is a Simple Finite Square Well Problem?

A Simple Finite Square Well Problem is a theoretical model used in quantum mechanics to describe the behavior of a particle trapped inside a square potential well. The well is finite in size and has a constant potential within its boundaries, while the particle is assumed to have a constant energy.

2. What is the significance of studying a Simple Finite Square Well Problem?

Studying a Simple Finite Square Well Problem allows us to gain a better understanding of the behavior of particles in confined spaces and how they interact with potential barriers. It also has practical applications in fields such as solid-state physics and nanotechnology.

3. How is a Simple Finite Square Well Problem solved?

The problem is typically solved using mathematical techniques such as the Schrödinger equation and boundary conditions. The resulting solution gives us information about the energy levels and wave functions of the particle within the well.

4. Can a Simple Finite Square Well Problem be extended to more complex systems?

Yes, the Simple Finite Square Well Problem can be extended to more complex systems by adding additional potential barriers or changing the shape of the well. This allows for a more accurate representation of real-world scenarios and can provide insights into the behavior of particles in different environments.

5. What are some real-world applications of the Simple Finite Square Well Problem?

The Simple Finite Square Well Problem has applications in various fields, including semiconductor devices, quantum computing, and optical fibers. It is also used in the study of nuclear physics and can help us understand the behavior of particles in atomic nuclei.

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