Lecture 9 Feb 1#
Discussion: wavefunctions and energy levels of hydrogen
We know how \(u(r)\) behaves in the \(r\rightarrow 0\) and \(r\rightarrow\infty\) limits. We can use this to sketch the possible radial wavefunctions for different values of \(\ell\).
Now consider the ground state (\(n=1\)). What will the wavefunction look like, ie. what will be the \(\ell\) value and the shape of the radial wavefunction?
What about the first excited state (\(n=2\))? Second excited state (\(n=3\))?
What is the degeneracy? (How many states are there as a function of \(n\)?)
Important points:
The radial wavefunctions \(R(r)\) start off as \(\propto r^\ell\) at \(r=0\) and then at large \(r\) have to decay exponentially. In between they can have \(n_r=0,1,2,\dots\) radial nodes.
The ground state (\(n=1\)) has no radial nodes and \(\ell=0\), this is the 1s state. The wavefunction \(R(r)\) is just an exponential function of \(r\).
For the first excited state (\(n=2\)), we can either add a node in the angular direction \(\ell=1\) and keep \(n_r=0\), or we can add a node in the radial direction \(n_r=1\) and keep \(\ell=0\). These are the 2s and 2p states.
For the next excited state (\(n=3\)), we now have two nodes to distribute between the radial and angular directions, so we can have \((n_r,\ell)=(2,0), (1,1)\), or \((0,2)\). These are the 3s, 3p, 3d states.
Counting up all the possible states (including the \(2\ell+1\) different \(m\) values for each \(\ell\)) gives a total number of states corresponding to each energy level \(n\) of \(n^2\). (We’re ignoring spin here, once we take spin into account this becomes \(2n^2\) since each state can have either a spin up or spin down electron). There is a high degree of degeneracy (which comes from the high degree of symmetry of our Hamiltonian).
The effective potential \(V_\mathrm{eff}(r)\) is the sum of the repulsive centrifugal term \(\ell(\ell+1)\hbar^2/2\mu\) and the attractive Coulomb term \(-e^2/4\pi\epsilon_0 r\). For \(\ell>0\) there is a potential barrier preventing the particle from reaching \(r=0\), and we see that in the behavior of the wavefunction near the origin, which becomes flatter at \(r=0\) as \(\ell\) increases (\(\propto r^\ell\)).
The classically-forbidden region is the region where \(E<V(r)\) and the wave cannot propagate (wiggle) and instead evanesces (decays). (The same as in solutions to the 1D Schrödinger equation, e.g. finite square well).
Further reading#
Townsend section 10.2.