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Restrictions on Initial Values for the first derivatives of a wavefunction, for a bound state in the time independent Schrödinger Equation?

Physics Asked on March 14, 2021

The time independent wave function for a bound state given some potential function $V(r)$ is given by the time independent Schrödinger Equation

$$EPsi=-frac{hbar^2}{2m}left(frac{partial^2Psi}{partial{x^2}}+frac{partial^2Psi}{partial{y^2}}+frac{partial^2Psi}{partial{z^2}}right)+VPsi$$

One example of a wavefunction that is in a bound state would be the wavefunction of an electron in a hydrogen atom. For the Hydrogen Atom when $l$ and $m$ are both $0$ the wavefunction is spherically symmetric, and for a spherically symmetric wavefunction in a bound state the time independent Schrödinger Equation reduces to

$$EPsi=-frac{hbar^2}{2m}left(frac{partial^2Psi}{partial{r^2}}+frac{2}{r}frac{partialPsi}{partial{r}}right)+VPsi$$

and for a second order differential equation the initial value of a function, and the initial value of the functions derivative are needed for a unique solution to the differential equation. In the case of a spherically symmetric wave function an additional requirement to following the Schrödinger Equation is that the integral of the square of the wavefunction from $0$ to $infty$ must be finite, non zero, and converge. This puts restrictions on the initial values for $frac{partialPsi}{partial{r}}$ as not all initial values will satisfy the second condition given the initial value for $Psi$.

In the case of the electron in a hydrogen atom $Vproptofrac{1}{r}$ and there are analytical solutions to the wavefunction for an electron in a hydrogen atom.

For most potential functions $V(r)$ there are no analytical solutions to the wavefunction, and also no analytical solutions for finding the energy levels. This means that in general the wavefunction must be modeled numerically, and the Energy levels must also be approximated numerically.

I understand that in the case of a hydrogen atom $frac{partialPsi}{partial{r}}$ is not $0$, and for the ground state there is no location, in which it would be $0$, but for $n>1$ there are points, there are values of $r$, for which $frac{partialPsi}{partial{r}}=0$.

When the wavefunction for a bound state cannot be found analytically can the value for $frac{partialPsi}{partial{r}}$ at $r=0$, or the values for $r$ in which $frac{partialPsi}{partial{r}}=0$ be found analytically? If not would approximating these values be similar to approximating the values for the Energy Levels?

One Answer

Usually the BC are not on the derivative of $psi$ but on $psi$ itself. For hydrogen, $lim_{rto 0}r^2 psi^2(r)to 0$ and $lim_{rtoinfty}psi(r)to 0$. The prob. density must have a node at $r=0$ by continuity since $r<0$ is not physical.

In practice the condition $psi(r)to 0$ as $rto infty$ is very difficult to implement numerically because of (unavoidable) roundoff errors: quantization occurs because the eigenvalue is exact, else the series for the differential equation does not exactly truncate and eventually diverges. Thus, the solutions are extremely sensitive to the guess energy and the accuracy of the integration scheme: even guess energy within 0.1% of the actual value will eventually produce divergences. In practice one chooses some “reasonably far” value of $r$ and looks for non-diverging solutions up to that point. It’s a bit of an art.

Answered by ZeroTheHero on March 14, 2021

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