Solving 1-D Potential TISE Homework: Get Solutions & Calculate Energies

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In summary, the conversation discusses a particle of mass m and energy E<V0 trapped in a 1-D potential well. The time independent Schrodinger equation is used to obtain solutions for the three regions, and by matching these solutions at the boundaries, the allowed energies can be found using the transcendental equation k cot(kL)=-\kappa. The most general solution for the differential equation in region 2 is given by \psi(x)=Asin(kx)+Bcos(kx), and this is a useful equation to know in physics.
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Homework Statement


A particle of mass m and energy E<V0 is trapped in the 1-D potential well defined by,
V=[tex]\infty[/tex], x<0; V=0, 0[tex]\leq[/tex]x[tex]\leq[/tex]L; V=V0, x>L.

(b) Obtain solutions to the time independent schrodinger equation for the three regions and by appropriate matching at the boundaries, show that the allowed energies are given by the transcendental equation, [tex]k cot(kL)=-\kappa[/tex], [tex]where[/tex][tex] {k^2}=\frac{2mE}{\hbar^2},{\kappa^2}=\frac{2m({V_0}-E)}{\hbar^2}[/tex].


Homework Equations


TISE


The Attempt at a Solution


For region 2,

[tex]\frac{{\partial^2}\psi}{\partial{x^2}}+\left(\frac{2mE}{\hbar^2}\right)\psi=0[/tex]

[tex]\frac{{\partial^2}\psi}{\partial{x^2}}+{k^2}\psi=0[/tex]

[tex]\psi(x)=Asin(kx)+Bcos(kx)[/tex]

I don't understand how to get from line two to line 3, and where does the potential term disappear to?
 
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8614smith said:
For region 2,

[tex]\frac{{\partial^2}\psi}{\partial{x^2}}+\left(\frac{2mE}{\hbar^2}\right)\psi=0[/tex]

[tex]\frac{{\partial^2}\psi}{\partial{x^2}}+{k^2}\psi=0[/tex]

[tex]\psi(x)=Asin(kx)+Bcos(kx)[/tex]

I don't understand how to get from line two to line 3, and where does the potential term disappear to?
Remember what the value of the potential is in region 2...

As for how to get from line 2 to line 3: first of all, I'll tell you directly that
[tex]\psi(x)=Asin(kx)+Bcos(kx)[/tex]
is the most general possible solution to the differential equation
[tex]\frac{{\partial^2}\psi}{\partial{x^2}}+{k^2}\psi=0[/tex]
This is a very useful thing for a physicist to know, because that equation comes up very frequently in physics. You're not going to want to go through the whole procedure of solving the equation every time you see it; it's awfully convenient to just say "the solution of [that equation] is..."

Since I gather that you're new to this equation, you should convince yourself that the solution is valid. The simple way is to just plug [itex]\psi(x)[/itex] into the equation and simplify it; you should find that everything cancels out, so the equation is satisfied. Showing that there are no other solutions is a little more complicated, and to learn about that I'd suggest taking a class on differential equations, if you have the opportunity. Otherwise, I'm sure there are good resources somewhere out there on the internet - a Google search or something should get you started.
 

1. What is the purpose of solving 1-D potential TISE homework?

The purpose of solving 1-D potential TISE (Time Independent Schrodinger Equation) homework is to understand and apply the fundamental principles of quantum mechanics to solve problems related to one-dimensional systems. This involves calculating the energies of the system and finding the corresponding wavefunctions.

2. How do I get solutions for 1-D potential TISE homework?

You can get solutions for 1-D potential TISE homework by first identifying the potential function and boundary conditions of the system. Then, you can use mathematical techniques such as separation of variables and boundary conditions to solve the TISE and obtain the wavefunction. Finally, you can use the wavefunction to calculate the energies of the system.

3. What are the different methods for solving 1-D potential TISE problems?

There are several methods for solving 1-D potential TISE problems, including the analytical method, numerical method, and approximation methods such as the variational method and perturbation theory. Each method has its own advantages and is suitable for different types of potential functions and boundary conditions.

4. Can I use a calculator to solve 1-D potential TISE homework?

Yes, you can use a calculator to solve 1-D potential TISE homework, but it is recommended to first understand the mathematical concepts and steps involved in the solution process. Additionally, some problems may require more advanced mathematical techniques that cannot be solved with a calculator.

5. How can I check if my solutions for 1-D potential TISE homework are correct?

You can check if your solutions for 1-D potential TISE homework are correct by comparing them with known results or by verifying that they satisfy the TISE and boundary conditions. Additionally, you can also use online tools or consult with a physics tutor for further clarification and verification of your solutions.

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