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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.
In quantum mechanics, the results of the quantum particle in a box can be used to look at the equilibrium situation for a quantum ideal gas in a box which is a box containing a large number of molecules which do not interact with each other except for instantaneous thermalizing collisions.
For the region outside of the box, since the potential is constant, () = and equation 1 becomes: = () There are two possible families of solutions, depending on whether E is less than V 0 {\displaystyle V_{0}} (the particle is in a bound state) or E is greater than V 0 {\displaystyle V_{0}} (the particle is in an unbounded state).
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.
This contrasts the solutions to position space wave functions, the position coordinates being continuous degrees of freedom, because then the Schrödinger equation does take the form of a wave equation. More generally, for a particle in 3d with any spin, the wave function can be written in "position–spin space" as: (,,) and these can also be ...
Consider Maxwell's set-up, but with only a single gas particle in a box. If the demon knows which half of the box the particle is in (equivalent to a single bit of information), it can close a shutter between the two halves of the box, close a piston unopposed into the empty half of the box, and then extract k B T ln 2 {\displaystyle k ...
What a twisted decision. A vagrant with 19 busts under his belt allegedly stabbed a 15-year-old boy with a corkscrew in an unprovoked Brooklyn subway attack — but the judge cut him loose without ...
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 ...