2P wavefunction
The term "2P wavefunction" typically refers to the wavefunction of an electron in a hydrogen-like atom (an atom with only one electron) when the electron occupies a 2p orbital. In quantum mechanics, the wavefunction describes the probability amplitude of finding a particle in a particular state.
The electron configuration for a hydrogen-like atom is often represented using the principal quantum number (n), azimuthal quantum number (l), and magnetic quantum number (m). For a 2p orbital:
Principal quantum number (n) = 2
Azimuthal quantum number (l) = 1
Magnetic quantum number (m) can take values from -1 to 1 (including 0)
The 2p orbital has three degenerate orbitals, often denoted as 2p_x, 2p_y, and 2p_z. Each of these orbitals corresponds to a specific set of quantum numbers, and the wavefunction for a general 2p orbital can be expressed as a linear combination of these individual orbitals.
It's important to note that the exact mathematical expression for the wavefunction depends on the coordinate system chosen and the specific quantum chemistry formalism used. The wavefunctions are often represented using spherical harmonics and radial functions.
The 2p wavefunction is a solution to the Schrödinger equation for the hydrogen atom or hydrogen-like ions, describing the behaviour of the electron in these systems. These wavefunctions are fundamental in understanding the electronic structure of atoms and molecules in quantum chemistry.
The 2p wavefunction is not spherically symmetric. A general method is to note that 1 / l r1 - r2 l is the generating function for the legendre polynomials.
Fig splitting of the first excited states of helium
where 𝜃_12 is the angle between r_1 and r_2. Then the angular parts of the integration are
As the ground state is spherically symmetric
In equation 1, the inner integral gives
Therefore, only the λ = 0 term contribute then we using
where n is the principal quantum number, l is the azimuthal quantum number, m is the magnetic quantum number, r is the radial distance, and θ,ϕ are the polar and azimuthal angles, respectively. We obtain
For the inner integral in eq 2
Only for l = λ contributes we have
This note is a part of the Physics Repository.