Solve Schroedinger Equation with Mathematica DSolve for given potential

In summary, the problem involves solving the Schroedinger equation using DSolve in Mathematica for a potential that is infinite below z=0 and V=mgz for positive z. The attempt at a solution involved defining the potential using Piecewise and then using DSolve to solve the differential equation with two boundary conditions. The error given by Mathematica indicated the use of inverse functions and suggested using V(z)=mgz for z>0 instead.
  • #1

Homework Statement


I need to solve the Schroedinger equation (Using DSolve in Mathematica) for a potential that is infinite below z=0 and V=mgz for positive z.


Homework Equations


TISE:
[itex] \psi \text{''}[z]+\frac{2 m}{\hbar }(\text{En}-V[z])\psi [z]==0 [/itex]



The Attempt at a Solution


First I defined the potential as
[itex]V[z]=\text{Piecewise}[\{\{\infty ,z<0\},\{m g z,z\geq 0\}\}];[/itex]

Then I told Mathematica to solve the Diffeq
[itex]\text{DSolve}\left[\left\{\psi \text{''}[z]+\frac{2 m}{\hbar }(\text{En}-V[z])\psi [z]==0,\psi [0]==0,\psi [\infty ]==0\right\},\psi [z],z\right][/itex]

The error mathematica gave me was
InverseFunction::ifun: Inverse functions are being used. Values may be lost for multivalued inverses. >>

I'm not really sure how to get mathematica to give me the solution. I know I need 2 conditions besides the differential equation in order to solve. The only boundary condition that I know to use is that the wavefunction must be zero at z=0 since the potential is infinite there.

Any ideas on what second boundary condition I should use or any ideas on where I'm going wrong? Thanks
 
Last edited:
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  • #2
It might be choking, in part, on the infinite potential. You know the wavefunction vanishes there, so try using V(z)=mgz and solve for just z>0.
 

What is the Schrödinger equation?

The Schrödinger equation is a fundamental equation in quantum mechanics that describes the time evolution of a quantum mechanical system. It was first proposed by Austrian physicist Erwin Schrödinger in 1926 and is essential for understanding the behavior of particles at the atomic and subatomic level.

What is the role of Mathematica in solving the Schrödinger equation?

Mathematica is a powerful computational software program that can be used to solve complex mathematical problems, including the Schrödinger equation. It has built-in functions and tools that make it easier to input and manipulate the necessary equations and parameters, allowing for faster and more accurate solutions.

What is the importance of solving the Schrödinger equation for given potentials?

Solving the Schrödinger equation allows us to determine the energy levels and wave functions of a quantum mechanical system. This is crucial for understanding the behavior of particles and predicting their interactions with other particles or external forces. By solving for specific potentials, we can also gain insight into the behavior of particles in different environments and conditions.

What are the limitations of using Mathematica to solve the Schrödinger equation?

While Mathematica is a powerful tool, it is not without its limitations when it comes to solving the Schrödinger equation. It may not be able to handle highly complex or nonlinear potentials, and the solutions may not always be accurate or physically meaningful. It is important to carefully choose the appropriate equations and parameters when using Mathematica to solve the Schrödinger equation.

Can Mathematica be used to solve the Schrödinger equation for systems with multiple particles?

Yes, Mathematica can be used to solve the Schrödinger equation for systems with multiple particles. However, as the number of particles increases, the complexity of the equations and the computational time required also increase significantly. In such cases, it may be more efficient to use other numerical methods or approximations to solve the Schrödinger equation.

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