SphericalpolarcoordinatesBefore proceeding, we recall the definition of spherical polar coordinates. They aredefined byx = r sindcos(py=rsinosingpz = r cosoand converselyr =/x?+y+??Q = arccos(= / r)β = arctan(y / x)The volume element for integration over spherical polar coordinates isdV=r?sinodrdodpForgetting the r?sino is a very common source of mistakes
Spherical polar coordinates Before proceeding, we recall the definition of spherical polar coordinates. They are defined by and conversely The volume element for integration over spherical polar coordinates is Forgetting the r 2 sinθ is a very common source of mistakes. sin cos sin sin cos x r y r z r 2 2 2 arccos / arctan / r x y z z r y x 2 d sin d d d V r r
aaaOra00adaxaxa0axadaraxOrOr2x=2rsin0cosd=sinOcosΦOxOxaa(sind+cotecosda0adaCiicoscotsin00Oda=-inInvolved only 0 and Φada2a1+cote-h000sin?a
y z y z y z , , , r x x r x x ˆ (sin cot cos ) x J i , 2 2 2 sin cos , sin cos y z r r r x r x x ˆ (cos cot sin ) y J i ˆ z J i 2 2 2 2 2 2 1 ˆ ( cot ) sin J Involved only and
V=-V(r)Acentral forceh?v? +V(r)2ma2P21f11Or.2forfrr sin'ofof0a22a1↑2Or2r?h?r Or[H,3,]=0H,j2=01aj2aj.+-,H)=0I.12,H)=0dtatatindtinh
A central force V =V(r) 2 2 ˆ ˆ ˆ ( ) 2 H T V V r m Ñ 2 = ¶ 2 ¶r 2 + 2 r ¶ ¶r + 1 r 2 ¶ 2 ¶q 2 + 1 r 2 cotq ¶ ¶q + 1 r 2 sin 2 q ¶ ¶f 2 2 2 2 2 2 2 1 ˆJ r r r r 2 ˆ ˆ H J, 0 z ˆ ˆ H J, 0 2 2 2 ˆ 1 ˆ ˆ [ , ] 0 d J J J H dt i t ˆ ˆ 1 ˆ ˆ [ , ] 0 z z z d J J J H dt i t
4.3SphericalharmonicsIn spherical polar coordinates, j, = -in %. J, commutes with j2, so we can find functions thatare eigenfunctions of both. Eigenfunctions of J, satisfy the eigenvalue equationj.y=-ih-y=kyaThe unnormalised solutions are of the form exp(ik/h), but the value of k is restricted by therequirement that the wavefunction is single-valued - that is, ( + 2r) must be the same as(p). This means thatexp(ik(β + 2元) / h) = exp(ikp / h)Thus k = Mh, where M is an integer (positive, negative or zero), and the wavefunctionsbecome(afternormalisation)1exp(iMp)VM2元Theeigenvalue is Mh :the angular momentum is an integer multiple ofh
In spherical polar coordinates, 𝐽መ 𝑧 = −𝑖ℏ 𝜕 𝜕𝜑. 𝐽መ 𝑧 commutes with 𝑱2 , so we can find functions that are eigenfunctions of both. Eigenfunctions of 𝐽መ 𝑧 satisfy the eigenvalue equation The unnormalised solutions are of the form exp(𝑖𝑘𝜑/ℏ), but the value of k is restricted by the requirement that the wavefunction is single-valued — that is, 𝜓(𝜑 + 2𝜋) must be the same as 𝜓(𝜑). This means that Thus 𝑘 = 𝑀ℏ, where M is an integer (positive, negative or zero), and the wavefunctions become (after normalisation) The eigenvalue is 𝑀ℏ : the angular momentum is an integer multiple of ℏ. ˆ z J i k exp 2 / exp / ik ik 1 exp 2 M iM 4.3 Spherical harmonics
Eigenfunctions of f2J2 is more complicated:aa2a112 -一sinsin? osingaeaoTo obtain eigenfunctions of f2 we have to multiply the functions eiMo by suitable functions of0. WriteYM = OM(0)M(β)where Φm(p) = eiMs, Then the eigenvalue equation J2 Yjm = AY jm becomesM?aaJ'Yjm =-h?sine0M(0)ΦM(p)=a0JM(0)Φ(p)singaoaosin?0We can cancel out Φm(p) to get an eigenvalue equation in 0. The eigenvalues turn out to beh?JU +1)for integerJ.The functions Y μm(0, β) are spherical harmonics
Eigenfunctions of 𝑱 𝟐 𝑱2 is more complicated: To obtain eigenfunctions of 𝑱2 we have to multiply the functions e 𝑖𝑀𝜑 by suitable functions of θ. Write where Φ𝑀(𝜑) = e 𝑖𝑀𝜑 . Then the eigenvalue equation 𝑱2𝑌𝐽𝑀 = 𝜆𝑌𝐽𝑀 becomes We can cancel out Φ𝑀(𝜑) to get an eigenvalue equation in θ. The eigenvalues λ turn out to be ℏ 2 𝐽(𝐽 + 1) for integer J. The functions 𝑌𝐽𝑀(𝜃,𝜑) are spherical harmonics. 2 2 2 2 2 1 1 ˆ sin sin sin J ( ) ( ) YJM JM M 2 2 2 2 1 ˆ sin ( ) ( ) ( ) ( ) sin sin JM JM M JM M M Y J