Calculating expectation value of angular kinetic energy

[TopCorN]

Homework Statement

the origial question given is: Show that the difference in energies between 2s and 2p radial wavefunctions is equal to the energy of the angular part of the 2p wavefunction, and thus that they have the same overall energy.

hints given:a)use virial theorem to determine total k.e. b)angular part is related to L(L+1)/r²

Homework Equations

I know: energy = -13.6z²/n² eV

angular k.e. = (h-bar)^2/(2m_er^2)*L(L+1)*R(r)

radial k.e. = -(h-bar)^2/(2me)*((1/r^2)*(d/dr)*(r^2)*(dR(r)/dr))

The Attempt at a Solution

today we were told that for this problem we need to calculate <E_k> for angular and radial energy.

my question is, what is the wavefunction i use for the expectation values?

I found in my textbook the wavefunction for 2p orbital with m_l = 0 is 1/(4*sqrt(2Pi))(Z/a₀)^(5/2)(r*cos(theta)*e^Zr/2a₀). is this what I am looking for


,thanks

 

[nate808]

for asking this question! It's always important to clarify what wavefunction to use when calculating expectation values.

In this case, you are correct that the wavefunction for the 2p orbital with ml = 0 is the one you should use. However, it's important to note that this is only one part of the overall wavefunction for the 2p orbital. The full wavefunction would also include the radial part, which is dependent on the distance from the nucleus.

When calculating expectation values, you need to use the full wavefunction, which includes both the angular and radial parts. So in this case, you would use the wavefunction you mentioned, but also include the radial part as well.

Hope that helps! Let me know if you have any other questions.