section_path: Examples/Particle in a box
section_name: Particle in a box
paragraph_text: Particle in a box
  
  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. For the one-dimensional case in the x {\displaystyle x} direction, the time-independent Schrödinger equation may be written
  
  With the differential operator defined by
  
  the previous equation is evocative of the classic kinetic energy analogue,
  
  with state ψ {\displaystyle \psi } in this case having energy E {\displaystyle E} coincident with the kinetic energy of the particle.
  
  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. Thus, at x = 0 {\displaystyle x=0},
  
  and D = 0 {\displaystyle D=0}. At Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle x=L},
  
  in which C {\displaystyle C} cannot be zero as this would conflict with the postulate that ψ {\displaystyle \psi } has norm 1. Therefore, since sin (k L) = 0 {\displaystyle \sin(kL)=0}, k L {\displaystyle kL} must be an integer multiple of π {\displaystyle \pi },
  
  This constraint on k {\displaystyle k} implies a constraint on the energy levels, yielding
  
  E n = ℏ 2 π 2 n 2 2 m L 2 = n 2 h 2 8 m L 2. {\displaystyle E_{n}={\frac {\hbar ^{2}\pi ^{2}n^{2}}{2mL^{2}}}={\frac {n^{2}h^{2}}{8mL^{2}}}.}
  
  A finite potential well is the generalization of the infinite potential well problem to potential wells having finite depth. The finite potential well problem is mathematically more complicated than the infinite particle-in-a-box problem as the wave function is not pinned to zero at the walls of the well. Instead, the wave function must satisfy more complicated mathematical boundary conditions as it is nonzero in regions outside the well. Another related problem is that of the rectangular potential barrier, which furnishes a model for the quantum tunneling effect that plays an important role in the performance of modern technologies such as flash memory and scanning tunneling microscopy.