Quantum particle in a box

In quantum mechanics, the particle in a box model (also known as the infinite potential well or the infinite square well) describes a particle free to move in a small space surrounded by impenetrable barriers. The model is mainly used as a hypothetical example to illustrate the differences between classical and quantum systems. In classical systems, for example a ball trapped inside a heavy box, the particle can move at any speed within the box and it is no more likely to be found at one position than another. However, when the well becomes very narrow (on the scale of a few nanometers), quantum effects become important. The particle may only occupy certain positive energy levels. Likewise, it can never have zero energy, meaning that the particle can never “sit still”. Additionally, it is more likely to be found at certain positions than at others, depending on its energy level. The particle may never be detected at certain positions, known as spatial nodes.

The particle in a box model provides one of the very few problems in quantum mechanics which can be solved analytically, without approximations. This means that the observable properties of the particle (such as its energy and position) are related to the mass of the particle and the width of the well by simple mathematical expressions. Due to its simplicity, the model allows insight into quantum effects without the need for complicated mathematics. It is one of the first quantum mechanics problems taught in undergraduate physics courses, and it is commonly used as an approximation for more complicated quantum systems. See also: the history of quantum mechanics.

One-dimensional solution

In quantum mechanics, the wavefunction gives the most fundamental description of the behavior of a particle; the measurable properties of the particle (such as its position, momentum and energy) may all be derived from the wavefunction.[3] The wavefunction ψ(x,t) can be found by solving the Schrödinger equationfor the system

\mathrm{i}\hbar\frac{\partial}{\partial t}\psi(x,t) = -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}\psi(x,t) +V(x)\psi(x,t),

where \hbar is the reduced Planck constantm is the mass of the particle, i is the imaginary unit and t is time.

Inside the box, no forces act upon the particle, which means that the part of the wavefunction inside the box oscillates through space and time with the same form as a free particle:[1][4]

\psi(x,t) = [A \sin(kx) + B \cos(kx)]\mathrm{e}^{-\mathrm{i}\omega t},\;

where A and B are arbitrary complex numbers. The frequency of the oscillations through space and time are given by the wavenumber k and the angular frequency ω respectively. These are both related to the total energy of the particle by the expression

E = \hbar\omega = \frac{\hbar^2 k^2}{2m},

which is known as the dispersion relation for a free particle.[1]

Initial wavefunctions for the first four states in a one-dimensional particle in a box

The size (or amplitude) of the wavefunction at a given position is related to the probability of finding a particle there by P(x,t) = | ψ(x,t) | 2. The wavefunction must therefore vanish everywhere beyond the edges of the box.[1][4] Also, the amplitude of the wavefunction may not “jump” abruptly from one point to the next.[1] These two conditions are only satisfied by wavefunctions with the form

\psi_n(x,t) = \begin{cases} A \sin(k_n x)\mathrm{e}^{-\mathrm{i}\omega_n t}, & 0 < x < L,\\ 0, & \text{otherwise,} \end{cases}

where n is a positive, whole number. The wavenumber is restricted to certain, specific values given by[5]

k_n = \frac{n \pi}{L}, \quad \mathrm{where} \quad n = \{1,2,3,4,\ldots\},

where L is the size of the box.[7] Negative values of n are neglected, since they give wavefunctions identical to the positive n solutions except for a physically unimportant sign change.[6]

Finally, the unknown constant A may be found by normalizing the wavefunction so that the total probability density of finding the particle in the system is 1. It follows that

\left| A \right| = \sqrt{\frac{2 }{L}}.

Thus, A may be any complex number with absolute value √(2/L); these different values of A yield the same physical state, so A = √(2/L) can be selected to simplify.

Energy levels

The energy of a particle in a box (black circles) and a free particle (grey line) both depend upon wavenumber in the same way. However, the particle in a box may only have certain, discrete energy levels.

The energies which correspond with each of the permitted wavenumbers may be written as[5]

E_n = \frac{n^2\hbar^2 \pi ^2}{2mL^2} = \frac{n^2 h^2}{8mL^2}.

The energy levels increase with n2, meaning that high energy levels are separated from each other by a greater amount than low energy levels are. The lowest possible energy for the particle (its zero-point energy) is found in state 1, which is given by[8]

E_1 = \frac{\hbar^2\pi^2}{2mL^2}.

The particle, therefore, always has a positive energy. This contrasts with classical systems, where the particle can have zero energy by resting motionless at the bottom of the box. This can be explained in terms of the uncertainty principle, which states that the product of the uncertainties in the position and momentum of a particle is limited by

\Delta x\Delta p \geq \frac{\hbar}{2}

It can be shown that the uncertainty in the position of the particle is proportional to the width of the box.[9] Thus, the uncertainty in momentum is roughly inversely proportional to the width of the box.[8] The kinetic energy of a particle is given by Ep2 / (2m), and hence the minimum kinetic energy of the particle in a box is inversely proportional to the mass and the square of the well width, in qualitative agreement with the calculation above.[8]

Spatial location

In classical physics, the particle can be detected anywhere in the box with equal probability. In quantum mechanics, however, the probability density for finding a particle at a given position is derived from the wavefunction as P(x) = | ψ(x) | 2. For the particle in a box, the probability density for finding the particle at a given position depends upon its state, and is given by

P_n(x) = \begin{cases}   \frac{2  }{L}\sin^2\left(\frac{n\pi x}{L}\right); & 0 < x < L \\   0; & \text{otherwise}. \end{cases}

Thus, for any value of n greater than one, there are regions within the box for which P(x) = 0, indicating that spatial nodes exist at which the particle cannot be found.

In quantum mechanics, the average, or expectation value of the position of a particle is given by

\langle x \rangle = \int_{-\infty}^{\infty} \psi^*(x) x \psi(x)\,\mathrm{d}x.

For the particle in a box, it can be shown that the average position is always \langle x \rangle = L/2, regardless of the state of the particle. In other words, the average position at which a particle in a box may be detected is exactly in the center of the quantum well; in agreement with a classical system.

Higher-dimensional boxes

If a particle is trapped in a two-dimensional box, it may freely move in the x and y-directions, between barriers separated by lengths Lx andLy respectively. Using a similar approach to that of the one-dimensional box, it can be shown that the wavefunctions and energies are given respectively by

\psi_{n_x,n_y} = \sqrt{\frac{4}{L_x L_y}} \sin \left( k_{n_x} x \right) \sin \left( k_{n_y} y\right),
E_{n_x,n_y} = \frac{\hbar^2 k_{n_x,n_y}^2}{2m},

where the two-dimensional wavevector is given by

\mathbf{k_{n_x,n_y}} = k_{n_x}\mathbf{\hat{x}} + k_{n_y}\mathbf{\hat{y}} = \frac{n_x \pi }{L_x} \mathbf{\hat{x}} + \frac{n_y \pi }{L_y} \mathbf{\hat{y}}.

For a three dimensional box, the solutions are

\psi_{n_x,n_y,n_z} = \sqrt{\frac{8}{L_x L_y L_z}} \sin \left( k_{n_x} x \right) \sin \left( k_{n_y} y \right) \sin \left( k_{n_z} z \right),
E_{n_x,n_y,n_z} = \frac{\hbar^2 k_{n_x,n_y,n_z}^2}{2m},

where the three-dimensional wavevector is given by

\mathbf{k_{n_x,n_y,n_z}} = k_{n_x}\mathbf{\hat{x}} + k_{n_y}\mathbf{\hat{y}} + k_{n_z}\mathbf{\hat{z}} = \frac{n_x \pi }{L_x} \mathbf{\hat{x}} + \frac{n_y \pi }{L_y} \mathbf{\hat{y}} + \frac{n_z \pi }{L_z} \mathbf{\hat{z}}.

An interesting feature of the above solutions is that when two or more of the lengths are the same (e.g. LxLy), there are multiple wavefunctions corresponding to the same total energy. For example the wavefunction with nx = 2,ny = 1 has the same energy as the wavefunction with nx = 1,ny = 2. This situation is called degeneracy and for the case where exactly two degenerate wavefunctions have the same energy that energy level is said to be doubly degenerate. Degeneracy results from symmetry in the system. For the above case two of the lengths are equal so the system is symmetric with respect to a 90° rotation.

References

http://en.wikipedia.org/wiki/Particle_in_a_box

http://en.wikipedia.org/wiki/Quantum_harmonic_oscillator

Modern Quantum Mechanics (2nd Edition)

Quantum Mechanics Non-Relativistic Theory, Third Edition: Volume 3

Introduction to Quantum Mechanics (2nd Edition)

Introductory Quantum Mechanics (4th Edition)

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