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Chapter 18: Particle in a Box

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18.1 Introduction to Particle in a Box

The particle in a box is a simplified model in quantum mechanics that helps to understand the behavior of a quantum particle confined in a one-dimensional, infinitely deep potential well. This model is important because it provides an elementary example of bound states, quantization of energy levels, and wavefunction normalization.

18.2 Time-Independent Schrödinger Equation

To analyze the particle in a box, we first consider the time-independent Schrödinger equation for a one-dimensional system:

-\dfrac{\hbar^2}{2m}\dfrac{d^2\psi}{dx^2} + V(x)\psi(x) = E\psi(x)

where \hbar is the reduced Planck’s constant, m is the mass of the particle, V(x) is the potential energy, \psi(x) is the wavefunction, and E is the total energy.

18.3 The Potential Well

For a particle in a box, the potential is given by:

V(x) = \begin{cases} 0, & 0 < x < L \\ \infty , & \text{otherwise} \end{cases}

This potential describes a particle confined between x=0 and x=L, and it is infinite outside this range. Therefore, the particle is constrained to move only within this region.

18.4 Bound States and Quantization of Energy

Solving the time-independent Schrödinger equation for the particle in a box, we obtain the wavefunction:

\psi_n(x) = \sqrt{\dfrac{2}{L}}\sin{\left(\dfrac{n\pi x}{L}\right)}

where n is an integer greater than zero, and L is the width of the box. The energy levels are quantized and can be expressed as:

E_n = \dfrac{n^2\pi^2\hbar^2}{2mL^2}

As n increases, the energy levels increase quadratically, indicating that only specific energy values are allowed for the particle confined in the box.

18.5 Normalization and Probability

Normalization ensures that the total probability of finding the particle in the box is equal to 1. The wavefunction for the particle in a box model is already normalized, as shown by the \sqrt{\frac{2}{L}} factor. The probability of finding the particle in a specific region is given by the square of the wavefunction:

P(x) = |\psi_n(x)|^2 = \dfrac{2}{L}\sin^2{\left(\dfrac{n\pi x}{L}\right)}

This probability distribution shows that the particle has a higher probability of being found in specific regions depending on the energy level, with nodes (points of zero probability) appearing at regular intervals.

Chapter Summary

The particle in a box model is a fundamental example in quantum mechanics that introduces the concepts of bound states, energy quantization, and wavefunction normalization. By solving the time-independent Schrödinger equation for this system, we can derive the quantized energy levels and probability distribution for a particle confined within a one-dimensional potential well. The particle in a box serves as a foundation for understanding more complex quantum systems and phenomena.

Continue to Chapter 19: The Harmonic Oscillator

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