Can the radial component of the wave function satisfy the radial equation?

• vst98
In summary, the radial components of the continuum electron wave function, Rkl(r), satisfies the radial equation given by {\left[\frac{-\hbar }{2m}\frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2\frac{\partial }{\partial r}\right)+\frac{\hbar ^2l(l+1)}{2mr^2}-\frac{Z e^2}{r}\right]R=E R}. The solution for Rkl(r) is given by Rkl(r)= C rl Exp[-ikr] F[i/k + l + 1,2l+1;2ikr], where C is everything that does not
vst98

Homework Statement

Show that radial components of the continuum electron wave function
satisfies the radial equation:

${\left[\frac{-\hbar }{2m}\frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2\frac{\partial }{\partial r}\right)+\frac{\hbar ^2l(l+1)}{2m r^2}-\frac{Z e^2}{r}\right]R=E R}$

where:
E=(k*hbar)2/2m
R=Rkl(r)

Homework Equations

so, the radial component R is given as:

${R_{\text{kl}}(r)=\frac{C_{\text{kl}}}{(2l+1)!}(2{kr})^l\text{Exp}[-\text{ikr}]*F(i/k+l+1,2l+1;2\text{ikr})}$

The Attempt at a Solution

First I have rewritten R as:

Rkl(r)= C rl Exp[-ikr] F[i/k + l + 1,2l+1;2ikr]

where in C is everything that does not depend on r.

Great, I was thinking this will be pretty straightforward to show.

My reasoning was to insert R in the above radial equation, take those derivatives and at the end I will be able the factorize R out, so that I'm left with something like
HR=AR where A is the energy.

It did not work out like that
After taking the derivatives I'm left with a mess and I don't see how to factorize R out.
I mean, derivatives of Exp[-ikr] give me back the same function (multiplied by constant) so I can factorize that out but rl and F give me problems after derivation.
Also I was looking at the properties of these confluent hypergeometric functions F
and tried to adjust them back to original function but without success.

Can someone say whether I'm on the right track trying to solve the problem like that or
have I completely missed the point ?

Interesting problem. I would first try to relabel the confluent hypergeom function as u(r) and plug the solution into the original ODE. Then I'd be led to a second order ODE for u(r) which should have exactly the same solution with the confluent hypergeom function I'm given.

I'm not sure if I understood you well, but this is what I have tried:

Now R is given as:

Rkl(r)= C rl Exp[-ikr] u[r]

And a second order ODE that u[r] obeys is:

$z\frac{d^{2}u}{dz^{^{2}}}+(b-z)\frac{du}{dz}-au=0$

where u is u[r]~F[a,b,z]=F[i/k+l+1,2l+1,2ikr]

So, I should insert R the into radial equation and with the help of this ODE for u[r] somehow simplify the result, right ?
Ok, I focused here only on the result I get when acting on R with this part of the radial equation
$\frac{1}{r^{2}}\frac{\partial }{\partial r}(r^{2}\frac{\partial }{\partial r})$

the result after acting on R with this part is:
${c e^{-i k r} r^{-2+l} \left(\left(l+l^2-2 i k l r-k r (2 i+k r)\right) u[r]+r \left(2 (1+l-i k r) u'[r]+r u''[r]\right)\right)}$

Now, I have tried to bring this expression(or a part of it) to form of ODE for u[r].
But when I adjust the coefficient that multiplies u''[r] so that I have 2ikr*u''[r] , which is required by the ODE for u[r] then the coefficient for u'[r] is not in a proper form anymore.
And the rest of the equation is also of no help since these are the only u''[r] and u'[r] terms.
So I'm stuck again :)

Last edited:

What is the radial component of the wave function?

The radial component of the wave function is a mathematical function that describes the variation of the wave in terms of distance from the center of the wave. It is an essential part of the wave function, which is a mathematical representation of a quantum system.

What is the radial equation?

The radial equation is a mathematical equation that describes the behavior of the radial component of the wave function. It is derived from the Schrödinger equation and is used to solve for the radial component of the wave function in quantum mechanics.

Why is it important to show that the radial component of the wave function satisfies the radial equation?

Showing that the radial component of the wave function satisfies the radial equation is important because it confirms that the wave function accurately describes the behavior of a quantum system. It also allows us to make predictions about the behavior of the system and understand its properties.

How do you show that the radial component of the wave function satisfies the radial equation?

To show that the radial component of the wave function satisfies the radial equation, we can substitute the wave function into the radial equation and then use mathematical techniques to simplify the equation and show that it is satisfied. This process involves manipulating the wave function using operators such as the Hamiltonian and the Laplacian.

What are the implications of the radial component of the wave function satisfying the radial equation?

The implication of the radial component of the wave function satisfying the radial equation is that it provides a complete and accurate description of the behavior of a quantum system. This allows us to make predictions about the system's properties and behavior, which is crucial in understanding and studying quantum mechanics.

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