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Some trajectories of a particle in a box according to Newton's laws of classical mechanics (A), and according to the Schrödinger equation of quantum mechanics (B–F). In (B–F), the horizontal axis is position, and the vertical axis is the real part (blue) and imaginary part (red) of the wave function.
The general solutions of the Schrödinger equation for the particle in a box are = + = or, from Euler's formula, = + (). The infinite potential walls of the box determine the values of C , D , {\displaystyle C,D,} and k {\displaystyle k} at x = 0 {\displaystyle x=0} and x = L {\displaystyle x=L} where ψ {\displaystyle \psi } must be zero.
The particle in a one-dimensional potential energy box is the most mathematically simple example where restraints lead to the quantization of energy levels. The box is defined as having zero potential energy everywhere inside a certain region, and therefore infinite potential energy everywhere outside that region.
The presence of degenerate energy levels is studied in the cases of Particle in a box and two-dimensional harmonic oscillator, which act as useful mathematical models for several real world systems. Particle in a rectangular plane
The wave function of the ground state of a particle in a one-dimensional box is a half-period sine wave, which goes to zero at the two edges of the well. The energy of the particle is given by h 2 n 2 8 m L 2 {\textstyle {\frac {h^{2}n^{2}}{8mL^{2}}}} , where h is the Planck constant , m is the mass of the particle, n is the energy state ( n ...
Note that exceptions can arise to the continuity of derivatives rule at points of infinite discontinuity of potential field. For example, in particle in a box where the derivative of wavefunction can be discontinuous at the boundary of the box where the potential is known to have infinite discontinuity.
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Some trajectories of a particle in a box according to Newton's laws of classical mechanics (A), and matter waves (B–F). In (B–F), the horizontal axis is position, and the vertical axis is the real part (blue) and imaginary part (red) of the wavefunction. The states (B,C,D) are energy eigenstates, but (E,F) are not.