Nuclear and Particle Physics SubAtomic Physics

Central symmetric potentialPotential energy function V=V(r) is function of r only, not of θ and ϕSolution of TISE:• radial wavefunction R(r) solution of radial TISE:2( l 1)⎡ h 1 d ⎛ 2 d ⎞ l + h⎢−r + V(r) +2µ2⎜ ⎟r dr dr2⎣ ⎝ ⎠ 2µr2centrifugal potential(or barrier)⎤⎥R( r) = ER( r)⎦• spherical harmonics Y(θ,ϕ) solution of angular TISE:⎡− ⎢⎣1sin θu(r,θ, φ) = R(r)Y(θ,φθ,φ)∂ ⎛ ∂ 1sin θ⎞⎜ ⎟ +∂θ⎝ ∂θ⎠ sin2∂2θ ∂φ2⎤⎥Y θ,φ⎦( ) = λY( θ,φ)Parity πParity operator ⇒ inversion in the origin coordinatesPˆ uIn polar coordinates:nlmm( r, θ, φ) = Pˆ R( r) Y ( θ,φ)r → rθ → π − θφ → π + φr r→ −Behaviour of eigenfunction under parity transformation determinedby properties of spherical harmonics.ml m[ ] = R( r) Y ( π − θ, π + φ) = R( r)( − 1) Y ( θ,φ)lPˆunlmEigenfunctions have definite parity:ll( r, θ,φ) = ( − 1) u ( r, θ,φ)nlmEVEN (or positive) for even lODD (or negative) for odd llN.B. potential V(r) does not appear in angular equation⇒always solutions, no matter what form for V(r)Ylm ( θ,φ)Very general result following from spherical symmetry of potentialm| ( θ,φ)Y l2|Distance of eachpoint from the originproportional to themagnitude in thatdirectionPolar plots of sections at y=0 through spherical harmonicszz+0 2| Y0 ( θ , φ)|Even parityy+-0 2| Y1 ( θ,φ)|Odd parityWe will see in later lectures how conservation of total angular momentum andparity are of crucial importance in understanding nuclear structure andreactions This overview deals with the QM of stationary states produced in astatic potential. To calculate the probability of transitions between thesestates we need more advanced QM. (We will touch a little on this later whenwe look at electron scattering -Fermi’s golden rule)y+-+ -0 2| Y2 ( θ,φ)|Even parity+, - sign offunction ineach region