# (Baby QM) Analytic Solution to the Infinite Square Well Problem

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• obstinatus
In summary, the individual is seeking to understand the 1-D time-independent Schrodinger's equation infinite square well problem and is questioning the analytical solution to plot against the normalized probability of finding the particle in a specific location. They mention the normalized wavefunction given by Hyperphysics being too large and welcome any suggestions. They also suggest using a nonlinear curve fit to find the correct multiplier for the probability density function.
obstinatus
Hi,

I think I'm having a bit of a brain fart...I'm messing with this numerical code trying to understand the 1-D time-independent Schrodinger's equation infinite square well problem (V(x) infinite at the boundaries, 0 everywhere else). If normalized Phi squared is the probability of finding the particle in that location, what the heck is the analytical solution I should plot against it to see how close it is? The normalized wavefunction given by Hyperphysics is much too large. Any and all suggestions appreciated.

obstinatus said:
Hi,

I think I'm having a bit of a brain fart...I'm messing with this numerical code trying to understand the 1-D time-independent Schrodinger's equation infinite square well problem (V(x) infinite at the boundaries, 0 everywhere else). If normalized Phi squared is the probability of finding the particle in that location, what the heck is the analytical solution I should plot against it to see how close it is? The normalized wavefunction given by Hyperphysics is much too large. Any and all suggestions appreciated.
View attachment 260814
What do you mean their wavefunction is too large? What specific wavefunction were you looking at?

That probability density function looks like it has to be multiplied with a constant greater than 50 to become normalized. Then you can compare it to the exact solution.

You can also find the correct multiplier by making a nonlinear curve fit to this data with function

##P (x) = C\sin^2 (2\pi x)##

and setting ##C## as the fitting parameter. This can be done in Origin Pro or some free program like Grace.

## 1. What is the Infinite Square Well Problem?

The Infinite Square Well Problem is a theoretical concept in quantum mechanics that describes a particle confined to a one-dimensional box with infinite potential energy barriers on either side. It is used as a simplified model to study the behavior of particles in a confined space.

## 2. What is the Baby QM Analytic Solution?

The Baby QM Analytic Solution is a mathematical solution to the Infinite Square Well Problem that uses the principles of quantum mechanics to determine the allowed energy levels and corresponding wave functions of a particle in the well. It is a simplified version of the full analytical solution, making it easier to understand and apply.

## 3. How is the Baby QM Analytic Solution derived?

The Baby QM Analytic Solution is derived by applying the Schrödinger equation, which describes the time evolution of a quantum system, to the Infinite Square Well Problem. This results in a set of differential equations that can be solved to determine the energy levels and wave functions of the particle in the well.

## 4. What are the limitations of the Baby QM Analytic Solution?

The Baby QM Analytic Solution is limited to one-dimensional systems and does not take into account factors such as spin and the effects of external forces. It also assumes that the potential energy is constant within the well, which may not always be the case in real-world systems.

## 5. How is the Baby QM Analytic Solution useful in real-world applications?

The Baby QM Analytic Solution serves as a starting point for understanding more complex quantum systems and can be applied to various physical systems, such as atoms, molecules, and semiconductors. It also provides insights into the behavior of particles in confined spaces, which is relevant in fields such as nanotechnology and quantum computing.

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