Potential with constant energy, and is the Reduction of the Effective Mass TDSE 2.1. In the final paragraph we present anĪpplication. We prove its correct simplification for constant mass and show that -normalizability Relates effective mass TDSEs and stationary Schrödinger equations to each Remainder of this note, we first give the point canonical transformation that Such that physical solutions are taken into physical solutions. Furthermore, our transformation preserves -normalizability, Of time-dependent potentials with effective masses and their corresponding This allows the straightforward generation Thus, each solvable stationary Schrödinger equation gives rise We identify a class of potentials for which the effective mass TDSE canīe reduced to a stationary Schrödinger equation by means of a point canonical The present note is to generalize the results in to the effective massĬase. Onto stationary problems has also been used for the calculation of Green'sįunctions for time-dependent Coulomb and other potentials. Most general class of reducible potentials is derived, particular cases haveīeen obtained earlier, for example, for time-dependent harmonic oscillator Stationary Schrödinger equation generates a solvable TDSE. Shown that for a certain class of potentials, the TDSE with constant mass canīe mapped onto a stationary Schrödinger equation, such that each solvable Order to attack this problem for noneffective (constant) mass, it has been However, the main problem of accessing time-dependent SchrödingerĮquations (TDSE) with effective mass is the lack of known solvable cases. These methods have also been elaborated for the fully time-dependent case Transformations (resp., supersymmetric factorization). Mainly by means of point canonical transformations and Darboux Stationary case, particular potentials with effective mass have been studied Since the effective mass SchrödingerĮquation takes a more complicated form than the conventional SchrödingerĮquation, the identification of solvable cases is more difficult. The quantum dynamics of suchĮlectrons can be modeled by an effective mass, the behavior of which isĭetermined by the band curvature.
IntroductionĮquations with effective mass occur in the context of transport phenomena inĬrystals (e.g., semiconductors), where the electrons are not completely free,īut interact with the potential of the lattice. This reduction is done by a particular point canonical transformation which preserves -normalizability.
Position-dependent (effective) mass allows reduction to a stationary Schrödinger equation. And you could do the same thing with the heat equation.We introduce a class of potentials for which the time-dependent Schrödinger equation with Then when you write the corresponding (linear) combination of separable solutions you get a solution to the time dependent equation that matches your initial conditions.Īnd often, that's all you really want. If you take your initial conditions, then you can write is as a (linear) combination of solutions to the time independent equation. What about solutions which are not separable? Why? Because then you can solve the separable equations as you describe by putting in a particularly simple time dependency. We are not requiring that $\psi$ be independent of time and be a solution to $H\psi=i\hbar\partial_t \psi.$ We require it be time independent and a solution to a completely new and different equation, $H\psi=E \psi.$ That is using the time dependent equation to look for particular solutions to the tine dependent equation that happen to be time independent. That is looking for an equilibrium or steady state. That is not what a time independent equation means.
If we allow $\Psi$ to be independent of time (which is my interpretation of a 'time independent equation') then why don't we just get $H\Psi=0$? Just like how most functions are not a solution to the heat equation. If you took a nonzero time independent solution to $H\psi=E\psi$ with nonzero energy $E$ then you'd notice right away that $H\psi=E\psi \neq0=i\hbar\partial_t \psi$ which means that function is simply not a solution to the time dependent equation. Its meaning is to take multiple spatial derivatives and do some other stuff. The left hand side has a meaning and the meaning isn't to take a single time derivative. To understand what is going on you have you to distinguish a definition from an equation.Īs an example you could consider the heat equation $\partial_$$