4.1: Infinite Potential Well
( \newcommand{\kernel}{\mathrm{null}\,}\)
Consider a particle of mass m and energy E moving in the following simple potential: V(x)={0for 0≤x≤a∞otherwise.
The solution to Equation ([e5.5]), subject to the boundary conditions ([e5.4]), is ψn(x)=Ansin(knx),
In other words, the eigenvalues of the energy operator are discrete. This is a general feature of bounded solutions: that is, solutions for which |ψ|→0 as |x|→∞. According to the discussion in Section [sstat], we expect the stationary eigenfunctions ψn(x) to satisfy the orthonormality constraint
∫a0ψn(x)ψm(x)dx=δnm.
Finally, again from Section [sstat], the general time-dependent solution can be written as a linear superposition of stationary solutions:
ψ(x,t)=∑n=0,∞cnψn(x)e−iEnt/ℏ,
Contributors and Attributions
Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)