Three-dimensional systems with spherical symmetry

Transcription

1 Thee-dimensiona systems with spheica symmety Thee-dimensiona systems with spheica symmety 006 Quantum Mechanics Pof. Y. F. Chen

2 Thee-dimensiona systems with spheica symmety We conside a patice moving in a centa potentia V that depends ony on the sepaation of the patice fom the cente of the potentia, and not on the spheica poa anges q and f that detemine paticua diections in space. The foce confining an eecton in an atom is the Couomb attaction to the nuceus, and the Couomb inteaction between two chaged patices is descibed by a centa potentia, vaying as /. 006 Quantum Mechanics Pof. Y. F. Chen

3 Thee-dimensiona systems with spheica symmety 006 Quantum Mechanics Pof. Y. F. Chen The Schödinge equation fo a patice moving in a spheicay symmetic potentia is given by 0. whee 0. With 0. and 0., the Schödinge equation can be witten in the fom 0.3 E V M v v h ψ ψ + ˆ sin sin sin L h + + φ,,,, ˆ φ ψ φ ψ E V L M + h h

4 Thee-dimensiona systems with spheica symmety Obita angua momentum opeatos commute with the Hamitonian, and the patice can be in a state that is simutaneousy an eigenstate of both enegy and angua momentum. Theefoe, we can assume that the enegy eigenfunctions of 0.3 ae of the fom ψ,, φ R Y, φ m Quantum Mechanics Pof. Y. F. Chen

5 Thee-dimensiona systems with spheica symmety Remembe that the eigenvaues of ˆL and Lˆ ae + and m, espectivey. The aowed vaues of the quantum numbe ae z 0,,,K 0.5 and, fo a given vaue of, the aowed eigenvaues of m ae m, +, +, L,,, 0.6 A wave function of 0.4 is an eigenfunction of eigenvaue +. ˆL coesponding to the Hence, substituting 0.4 into 0.3, we obtain the adia equation h M d d + + d d + V R E R Quantum Mechanics Pof. Y. F. Chen

6 Thee-dimensiona systems with spheica symmety The adia wave function R that descibes a bound state needs to be finite fo a, and ony fo a paticua set of discete enegy eigenvaues can exist. Since appeas in 0.7 but m does not, the adia wave function R and the enegy eigenvaue E depend ony on the eigenvaue of that of Lˆ. z ˆL but not on 006 Quantum Mechanics Pof. Y. F. Chen

7 Thee-dimensiona systems with spheica symmety The Hydogen atom The pobem fo the Couomb potentia is vey impotant because it foms the basis fo a discussions of atomic stuctue. It povides a pope quantum mechanica deivation of Boh s expession fo the enegy eves of the hydogen atom. The hydogen atom consists of an eecton and a poton inteacting though the potentia e V whee is the sepaation of the two patices Quantum Mechanics Pof. Y. F. Chen

8 Thee-dimensiona systems with spheica symmety The Hydogen atom The tota enegy can be sepaated into two pats, one associated with the motion of the cente of mass of the system and the othe with the eative motion of the two patices, and ony the second pat is of inteest in studying the intena stuctue of the two-patice system. Hence, the two-patice pobem is effectivey educed to a one-patice pobem with the potentia enegy given by 0.8 and a kinetic enegy opeato of the fom, h / μ, whee the educed mass μ is given by 0.9 me m p μ and m + m ae the masses of the fee eecton and poton, espectivey. me e m p p 006 Quantum Mechanics Pof. Y. F. Chen

9 Thee-dimensiona systems with spheica symmety The Hydogen atom The mass of the poton is about 836 times the mass of the eecton; the educed mass μ is not vey diffeent fom the eecton mass m. The sight diffeence between them has an expeimentay measuabe effect on the fequencies of the specta ines of hydogen, which confims that it is necessay to use the educed mass. Substituting 0.8 and 0.9 into 0.7 e h d d + + μ d d e R E R 0.0 It is convenient to define the dimensioness vaiabes: ρ, ε n a E B E R Quantum Mechanics Pof. Y. F. Chen

10 Thee-dimensiona systems with spheica symmety The Hydogen atom The Boh adius h ab, E μ e a B R and the Rydbeg enegy E R ae given by 4 e μe a h B In tems of ρ and ε, the adia wave equation 0.0 d d ρ d + n ε n ~ R ρ ρ d ρ ρ ρ With the esut of section 8.5 fo the associated Laguee poynomias, the eigenvaues of 0.3 can be found to be ε n ER En, n n,, Quantum Mechanics Pof. Y. F. Chen

11 Thee-dimensiona systems with spheica symmety The Hydogen atom The eigenfunction coesponding to the eigenvaue and angua E n momentum quantum numbe is given by ~ ρ / + R R ρ C e ρ L n, n, n, n ρ 0.5 whee C n, is the nomaization constant. Fo the dimensioness paamete ρ, the nomaization constant C n, is given by C n, n n! [ n +! ] Fo the adia vaiabe, the nomaized adia wave function is given by 3/ n! / n a + B R n, e L n ab n [ ] 3 n n +! n ab n ab Quantum Mechanics Pof. Y. F. Chen