Lectue 7: Angula Momentum, Hydogen Atom Vecto Quantization of Angula Momentum and Nomalization of 3D Rigid Roto wavefunctions Conside l, so L 2 2 2. Thus, we have L 2. Thee ae thee possibilities fo L z depending on the value of m l. We have L z 0,o L z o L z. Fo all of these, the length of angula momentum vecto is 2. Thus, we have a vecto whose length is 2 and whose pojection on the Z-axis can take only 3 possible values. Thus, the angula momentum vecto can eithe be in the X-Y plane o, anywhee on cones at angles of ±45 degees to the Z-axis. This is efeed to as vecto quantization of angula momentum. Similaly, we can wok out fo the l 2 states. In ode to nomalize the eigenfunctions of the igid oto Hamiltonian, we need to do an integal ove the angles θ and φ. The convesion of the volume integal in 3D fom Catesian to spheical pola coodinates involves a Jacobian of tansfomation so that we have dxdydz 2 sin(θ)ddθdφ Thus the nomalization condition fo ψ(θ, φ) is given as 2π π ψ (θ, φ)ψ(θ, φ) sin(θ)dθdφ 0 0
2 The Hydogen Atom Explaining the stability of the Hydogen atom is one of the tiumphs of the quantum theoy. Teating the H-atom as a poton of mass m p at R and an electon of mass m e at, we can wite the Hamiltonian as Ĥ( R, ) 2 2 R 2 2 2m p 2m e 4πɛ e 2 R Using a change of coodinates to cente of mass and elative coodinates, we can wite fo the elative motion Ĥ( ) 2 2µ 2 + e 2 4πɛ whee the Laplacian is with espect to the elative coodinate and is the distance between the electon and the nucleus. The educed mass µ is given by µ m p + m e m e. In othe wods, we can appoximate this system as a fixed nucleus and an electon obiting about it. The Laplacian can be witten in spheical pola coodinates and now the potential depends only on and not the angles. Thus we can wite the Hamiltonian opeato as Ĥ(, θ, φ) 2 2µ ( 2 2 + 2 The Schödinge equation is solved by sepaation of vaiables + ( 2 sin 2 θ 2 φ + 2 sin θ Ĥ(, θ, φ)ψ(, θ, φ) Eψ(, θ, φ) ψ(, θ, φ) R()Y (θ, φ) θ sin θ )) e 2 θ 4πɛ. The angula pat is exactly the spheical hamonics discussed in the igid oto poblem. The adial pat is denoted by R() and is dependent on two quantumnumbes, the pincipal quantum numbe n and the azimuthal 2
quantum numbe l, which also appeas in the spheical hamonics. Thus we have ψ n,l,ml (, θ, φ) R n,l ()Y l,ml (θ, φ) We mention hee the fom of R n,l (). You ae not expected to emembe this but it helps to undestand the key pats of the wavefunction. Fo convenience we define ρ 2/n whee h2 ɛ 0 πm e e 2 is efeed to as the Boh adius and is equal to 0.529 Å. The adial pat of the wavefunction can now be witten as R n,l () N n,l ρ l L n,l (ρ)e ρ/2 whee L n,l (ρ) is efeed to as the associated Laguee polynomial, a polynomial whose degee is n and N n,l is the nomalization constant. The fist few wavefunctions ae given below: n, l 0, s obital ψ,0,0 R,0 ()Y 0,0 (θ, φ) 2 e / 4π n 2, l 0, 2s obital ψ 2,0,0 R 2,0 ()Y 0,0 (θ, φ) ( ) 3/2 ) ( (2 a0 e /2 8 /2 4π ) /2 n 2, l, m l 0, 2p z obital ψ 2,,0 R 2, ()Y,0 (θ, φ) e /2 3 cos θ 24 /2 4π n 2, l, m l, 2p obital ψ 2,,0 R 2, ()Y,0 (θ, φ) e /2 3 sin θe iφ 24 /2 4π 3
n 2, l, m l, 2p obital ψ 2,,0 R 2, ()Y,0 (θ, φ) e /2 3 sin θe iφ 24 /2 4π The last two ae combined to give the eal 2p x and 2p y obitals. The enegy of a state, howeve, depends only on the pincipal quantum numbe n and is given by E n m ee 4 8h 2 ɛ 2 0n 2 The estictions on the quantum numbes ae as follows: n, 2, 3,... l 0,, 2...n m l 0, ±... ± l and they ae efeed to as pincipal, azimuthal and magnetic quantum numbes. The wavefunction ψ n,l,m is the wavefunction of a single electon and is efeed to as an obital o a spatial obital. Plotting of wavefunctions is a vey poweful tool. Since the wavefunction depends on a 3D position of the electon, we pefe to plot the angula and adial pats sepaately. Since the angula pats have aleady been plotted befoe when we looked at otational states, we plot the fist few adial pats below. Notice that fo s and 2s, the wavefunction is maximum at the nucleus, but fo 2p the wavefunction is zeo at the nucleus. The coesponding plots of the squae of the adial pat can easily be plotted. 4
s 2s 2p 3s 5