The Schödinge Equation in Thee Dimensions Paticle in a Rigid Thee-Dimensional Box (Catesian Coodinates) To illustate the solution of the time-independent Schödinge equation (TISE) in thee dimensions, we stat with the simple poblem of a paticle in a igid box. This is the theedimensional vesion of the poblem of the paticle in a one-dimensional, igid box. In one dimension, the TISE is witten as d ψ ( x) + U ( x) ψ ( x) = Eψ ( x). m dx (1) In thee dimensions, the wave function will in geneal be a function of the thee spatial coodinates. In the Catesian y coodinate system, these coodinates ae x, y, and. Also, the potential enegy U will in geneal be a function of all 3 x d ψ coodinates. Now, in the 1-D TISE, the tem L m dx px kx can be identified with the kinetic enegy = of the L y m m d ψ paticle because = [ E U ] ψ. [Ty, fo example, the fee-paticle wave function m dx i( kx ωt ψ = Ae ).] In thee dimensions, the KE is(p x + p y + p ) / m, so we suspect that additional second deivative tems will be needed to epesent the additional kinetic enegy tems. In fact, the TISE in thee dimensions is witten as ψ ( x, y, ) ψ ( x, y, ) ψ ( x, y, ) + + U ( x, y, ) ψ ( x, y, ) Eψ ( x, y, ). + = () m x y ψ (x, y, ) The symbol epesents the patial deivative of ψ (x, y, ) with espect to x, which is x simply the deivative of ψ (x, y,) with espect to x with y and held constant. Thus, ifψ = x y, then ψ x = xy. In the paticle in a box poblem, U(x, y, ) = 0 inside the box and U = at the walls. 1 So, inside the box, the TISE becomes ψ ψ ψ + + Eψ. = (3) m x y L x 1 Inside the box, 0<x<L x, 0<y<L y, and 0<<L. (See figue of box above.) 1
To solve patial diffeential equations (the TISE in 3D is an example of these equations), one can employ the method of sepaation of vaiables. We wite ψ (x, y,) = X(x)Y (y)z(), (4) whee X is a function of x only, Y is a function of y only, and Z is a function of only. Substituting fo ψ in Eq. (3) yields d X d Y d Z Y ( y) Z( ) X ( x) Y ( ) X ( x) Y ( y) EX ( x) Y ( y) Z( ). m + + = dx dy d Note that the patial deivatives have disappeaed, since X, Y, and Z ae functions of one vaiable only. Dividing both sides by X ( x) Y ( y) Z( ) yields m 1 d X 1 d Y 1 d Z me + + =. X dx Y dy Z d (6) Fo a given solution, E is a constant. Futhe, note that the fist tem in the squae backets is a function of x only, the second tem is a function of y only, and the thid tem is a function of only. Fo Eq. (6) to be valid fo all values of x, y, and, each of the thee tems must be constant. If this wee not tue, and, fo example, all the tems wee vaiable, then if one held y and constant and changed x, the sum on the left hand side of the equation would change, violating the equation. Eq. (6) theefoe becomes thee sepaated odinay diffeential equations: and 1 d X X dx 1 d Y Y dy 1 d Z Z d (5) = k x, (7) = k y, (8) = k, (9) with me k + k + k =. (10) x y The sepaation constants ae witten as k x, k y, and k in analogy with the 1-D paticle in a box poblem. Eqs. (7), (8), and (9) ae identical to the equation obtained in the 1-D poblem and the bounday conditions ae the same. Fo example, X(x) = 0 at x = 0 and x = L x since the wave functions cannot penetate the wall. The bounday condition at x = 0 leads to X(x) = A 1 sin k x x. The bounday condition at x = L x leads to k x = n xπ, whee n x = 1,, Simila solutions ae obtained fo Y(y) and Z(). Hence, we find that L x
ψ (x, y,) = Asin n xπ x sin n yπ y sin n π, (11) L x L y L and fom Eq. (10), we find the enegy π n n x y n E = + + m L x L y L. (1) The integes n x, n y, and n ae called quantum numbes. These quantum numbes specify the quantum states of the system. Thus, we can label the wave functions and enegies accoding to the values of the quantum numbes: ψ nx n y n ; E nx n y n. The gound state enegy is E 111 π 1 1 1 m L x L y L = + +. (13) If the box is a cube, then L x = L y = L = L, and the enegy becomes π En ( ). xnyn = n x + n y + n ml (14) The gound state enegy is then E 3 π =. ml 111 Degeneacy Fo the next highe enegy level up fom the gound state, thee ae 3 distinct wave functions o quantum states of the cubical box that have this enegy:ψ 11, ψ 11, ψ 11. These quantum states ae distinct because thei pobability densities ae diffeent. (E.g., ψ 11 = A π x sin L sin π y L sin π L A π x sin π y L sin π L sin L = ψ 11.) The occuence of moe than one distinct quantum states (wave functions) with the same enegy is called degeneacy. In the case of the paticle in a igid, cubical box, the next-lowest enegy level is thee-fold degeneate. 3 π E11 = E11 = E11 =. (15) ml. Note that the gound state is non-degeneate. Futhe, the thee-fold degeneacy of the fistexcited state is emoved if L x L y L. Degeneacy is intimately connected to symmety: the 3
geate the symmety of the system, the highe the degeneacy of the quantum states. [See www.falstad.com fo simulations.] Contou Maps Contous epesent lines of equal pobability density. See the contou maps fo the -D igid box in the Taylo et al. textbook. The Thee Dimensional Cental-Foce Poblem Ultimately, ou aim is to solve the TISE fo an atomic system. The simplest atomic system is a one-electon atom in which a single electon is bound to the nucleus. In this case, the potential enegy U(x, y, ) = U() = kze. (16) The potential enegy is a function of only the adial distance. The coesponding foce is diected along e. Such a foce (in this case, the electostatic foce) is called a cental foce. +Ze Note that U is spheically symmetic since it depends only on. Thus, classically, in a one-electon atomic system in which no extenal foces act, one does not expect the enegy E to depend on the angula position of the electon, only on the adial distance fom the nucleus. Because of the spheical symmety, the solution to the TISE is tactable if we use spheical pola coodinates athe than Catesian coodinates. In the spheical coodinate system, the coodinates ae, θ, andφ, whee is the adial distance, θ is the pola angle, and φ is the aimuthal angle. Fo a spheically symmetic potential enegy U(), the TISE cannot be solved using sepaation of vaiables in Catesian coodinates. In spheical coodinates, howeve, the TISE is sepaable, and this is the solution we shall study. It tuns out that even though the TISE is sepaable in spheical coodinates, the solution is complicated and we shall not go into all the mathematical details of the solution. Rathe, we shall concentate on the physics. x y = cosθ sin φ In spheical coodinates, the TISE fo a spheically symmetic potential is given by 1 1 ψ 1 ψ ( ψ ) + sin θ U ( ) ψ Eψ, + + = µ sinθ θ θ sin θ φ φ θ P = cosθ y x = cosθ cosφ (17) 4
whee ψ ψ (,θ,φ)and the mass is taken to be the educed mass µ. [Recall that this accounts fo the slight motion of the nucleus.] Using sepaation of vaiables, we seek a solution of the fom ψ (,θ,φ) = R()Θ(θ)Φ(φ). (18) Substituting in the TISE and sepaating yields and 1 d Φ Φ dφ = m l, (19) 1 d dθ sinθ sinθ dθ dθ + l(l + 1) m l sin θ Θ = 0, (0) d d µ l( l + 1) ( R) U ( ) + E R = 0, µ whee in Eq. (19), the sepaation constant is m and in Eq. (0), the sepaation constant is l(l + 1). Note that neithe the θ no the φ equation contains U(). Thus, these equations ae valid fo all cental foce potential enegies. We can solve Eq. (19) easily. We fist ewite it as d Φ dφ + m l Φ = 0, the solution of which is (1) Φ = Ae im l (φ +δ ). () In Eq. (), A and δ ae integation constants. The bounday condition is that Ф has to be singlevalued, i.e., Φ(φ) = Φ(φ + πn), whee n is an intege. This means that i.e., Angula Momentum e i(π nm l ) = 1, (3) m l = 0, ± 1, ±,... (4) imlφ The angula wave function Φ = Ae eminds us of the 1-D fee-paticle wave function (time suppessed) ψ (x) = Be ikx ipx /, which we can wite as ψ ( x) = Be, whee p = k is the paticle's il / linea momentum. Similaly, we can wite Φ ( φ) = Ae φ, whee L = ml. is the -component of the angula momentum. The impotant thing to note is that L is quantied fo a paticle with angula wave function Ф, i.e., it can assume only discete values ml, whee m l = 0, ±1, ±, is the quantum numbe associated with L (magnetic quantum numbe). In sepaating the θ-dependent pat of the TISE, the sepaation constant was taken to be l(l+1). It tuns out that the solutions to the θ-equation, i.e., the Θ(θ), ae of two kinds. One kind emains finite fo all 0 θ π fo non-negative intege values of l and the othe kind blows up at θ = 0 and π. Thus, only in the case of the fist kind will physically meaningful solutions be obtained. The θ-equation is known as the associated Legende equation, and the physically acceptable 5
m solutions ae the associated Legende functions of the fist kind, P l l (θ). Note that these functions depend on both l and m l. In fact, the solutions impose the condition l m l. The physical intepetation of this is that the magnitude of the obital angula momentum L is quantied fo a paticle with angula wave function Θ(θ) : L = l( l + 1), (5) o, L = l( l + 1). ( l = 0, 1,,...) (6) The intege l is the obital angula momentum quantum numbe. As seen befoe, the - component of L is quantied accoding to L = m. ( m = l, l + 1,..., l 1, l) (7) Vecto Model l l To visualie the implications of angula momentum quantiation, a classical pictue, i.e., the vecto model, is quite useful. If θ is the pola angle that L makes with the -axis, then L cos ml ml θ = = =. L l( l + 1) l( l + 1) (8) Now, fo a given value of l, m l can assume l+1 values [l, l+1,, l 1, l]. Hence, θ can assume only discete values, i.e., only cetain oientations of L ae allowed. This phenomenon is called spatial o space quantiation. L = L = L = 0 L = L = L = ( + 1) = 5 Vecto Model fo l =. Note that the oientation of the -axis is completely abitay. Whateve diection we choose fo it, a paticle in a state specified by ψ (,θ,φ) = R()Θ(θ)Φ(φ) will have definite values of L and L as specified above. 6
The Heisenbeg uncetainty pinciple tells us that only one component of L can be known exactly. Fo the coodinate system that we have chosen, this component is L. Thus, the othe two components, L x and L y, ae completely uncetain. We can see this fom the uncetainty elation ΔL Δφ /. (9) If L is known exactly then ΔL = 0. The uncetainty elation indicates that Δφ must be completely unspecified, in othe wods, all values ae equally pobable. To show this indeteminacy in φ (o equivalently L x and L y ), the L vecto is shown as lying anywhee on a cone of half angle θ, all positions being equally pobable. [Pictue below fom Kane s Moden Physics.] L Ly L x Enegy Levels of the Hydogen Atom The adial equation contains U() and so its solution will povide infomation about the enegy E of the paticle. Fo convenience, we wite it again below: d µ l( l + 1) ( R) U ( ) E R 0. d + = µ (30) The equation involves l and so in geneal E will depend on l. The adial equation, on the othe hand, does not depend on m l, and so E is always independent of m l fo cental foce potential enegies. This is consistent with the classical notion that fo a cental potential enegy, the enegy is independent of the oientation of the obit because of the spheical symmety. Let us focus on the vey impotant case of the hydogen atom, in which U() = ke. (Hydogen atom) (31) See Physics Today, May 004, p. 40. 7
The esulting adial equation can be solved analytically, and fo physically meaningful boundstate solutions satisfying the equisite bounday conditions (R 0 as, which is equivalent to nomaliability), one finds quantied enegies: µ ( ke ) 1 13.6 ev E = =. (Hydogen atom) (3) n n The numbe n is a thid quantum numbe called the pincipal quantum numbe, which can assume the intege values n = 1,, 3,. The physically acceptable solutions to the adial equation also equie that l can have the values 3 l = 0,1,,..., n 1. (Hydogen atom) (33) The solutions to the adial equation ae the adial functions R nl (), which ae elated to the associated Laguee functions. Notice that E is independent of m l and l. The independence of l is a special chaacteistic of the 1/ potential enegy. We shall see that in the case of multielecton atoms in which the potential is appoximately cental but not 1/, E does in fact depend on l. It is also impotant to note that the enegy levels pedicted by the TISE ae exactly the same as those pedicted by the Boh model, which ae known to be coect fo the hydogen atom. The values of the quantum numbes n, l, and m l aose fom bounday conditions on the sepaated functions R nl, Θ lml, and Φ ml. The complete wave function fo the hydogen atom is then ψ nlm (,θ,ψ ) = R nl ()Θ ml (θ)φ ml (φ). (34) The poduct of Θ lml (θ) and Φ ml (φ) is called the spheical hamonics: Y lml (θ,φ) = Θ ml (θ)φ ml (φ). (35) Each quantum state is uniquely specified by the values of the thee quantum numbes. Atomic states ae labeled with a numbe-lette code which specifies the values of the quantum numbes n and l. This labeling is known as spectoscopic notation. The numbe in the code is the value of n. The lette that follows designates the value of l: l = 0 is designated by s (shap); l = 1 by p (pincipal), l = by d (diffuse); and l = 3 by f (fundamental). Fo highe values of l, the code continues alphabetically, e.g., l = 4 is designated by g, l = 5 by h, and so on. We shall use spectoscopic notation to label the wave functions coesponding to each quantum state of the hydogen atom and constuct an enegy-level diagam showing the odeing of the enegies of the quantum states. We stat with the gound state. In the gound state of the H atom, n = 1. This implies that l = 0 and m l = 0. Thus, the gound state is the 1s state. Note also that the gound state is nondegeneate, i.e., the degeneacy = 1, since l = m l = 0. 4 Fo n =, l can have two values: 0 and 1. Fo l = 0, m l = 0 and this is the s state. Fo l = 1, we have thee possible value of m l : -1, 0, +1. Hence, thee ae thee p states. We see that thee ae fou wave functions (quantum states) fo n 3 l has to be limited because fo a given enegy the angula momentum cannot be abitaily lage. 4 Note that L = 0 fo gound state which diffes fom Boh model pediction of L=, The TISE s value is suppoted by expeiment. Also, actually, thee is a degeneacy due to spin. 8
=. All fou states have the same enegy, since E is independent of l and m l. Hence, the enegy level fo n = (fist excited state) is fou-fold degeneate. Fo n = 3, we have one 3s state, thee 3p states and five 3d states [m l = -, -1, 0, +1, +]. Thus, the degeneacy of the n = 3 enegy level is 1+3+5 = 9. In geneal, the degeneacy of the n th enegy level is n. [Show pictue fom Taylo et al.] We shall now investigate the pobability densities of hydogenic wave functionsψ nlml (,θ,φ). Hydogen Atom Wave Functions Pobability Densities In thee dimensions, the pobability density is a pobability pe unit volume. Hence, the pobability of finding the electon in a volume element dv at position (, θ,φ ) is dp(,θ,φ) = ψ nlml dv = R nl () Θ lml (θ) Φ ml (φ) sinθddθdφ. (36) Notice that Φ ml (φ) = A e im lφ = A. (37) Hence, the pobability of finding the electon within dv at a specified position is always independent of φ. It is vey useful to examine the pobability that the electon is at a cetain distance fom the nucleus, egadless of the values of θ andφ. Moe pecisely, we wish to find the pobability that an electon is between and +d, i.e., within the volume of a thin spheical shell. To do this, we fist find the volume dv of the spheical shell and then multiply it by R nl (). Now fo a spheical shell, dv = 4π d. (38) Then, the equied pobability is given by P()d = R() 4π d, (39) whee is the adial pobability density. [Show pictue, Kane, p. 185] P() = 4π R() (40) Example: (Example 7.3, Kane) Pove that the most likely distance fom the oigin of an electon in a n =, l =1 (p) state of H atom is 4a B. 9
Solution: P() = 4π R 1. Since we wish to find the value of whee P() is maximied, we need to find dp() and set if equal to eo. Now P() = (const.) 4 e /a B. [The multiplicative constants ae not d impotant in solving dp() d = 0.] Thus dp() d = C 4 3 e /a B + 4 1 e /a B a B = 0. Thus, C 3 e /a 0 4 a = 0, i.e., = 0,, o = 4a B. Obviously, = 0 o do not give a 0 maximum, since P() = 0 at those values of. We conclude that P() is maximum when = 4a B. [Show pictue, p. 185, Kane (again); Eisbeg & Resnick.] In geneal, as n inceases, the position of the highest peak of P(), i.e., the most pobable adial distance fo the electon, moves to lage values. The electon is theefoe fathe fom the nucleus on aveage. This can also be seen by calculating the expectation value of : = [Show Figs. in Taylo et al. and Eisbeg & Resnick.] 0 P()d. (41) While and mp (most pobable) ae cetainly diffeent fo the vaious states coesponding to the same n (e.g., s, p), this vaiation is elatively small compaed to the vaiation in (and mp ) fo diffeent n. Hence, one can identify a spatial shell with each n. A shell epesents a ange of distances within which the electon is most likely to be found. [Show diagam.] Fo example, if the electon is in a quantum state fo which n =, it is vey likely to be found in the n = spatial shell. One can also specify enegy shells which ae goups of quantum states with simila enegies. In the H atom (and all one-electon atoms), enegy shells and spatial shells ae identical because of the degeneacy of the levels fo each n. Howeve, in multielecton atoms, E does depend on l quite stongly in some cases, and states with diffeent n can have simila enegies. Angula Pobability Density As emaked peviously, s states have l = 0 and m l = 0. Mathematically, this means the functions Θ m ( θ ) and Φ ( ) l m φ ae both constants, i.e., independent of θ and φ, espectively. Hence, l ψ n00 has only a adial dependence and so is spheically symmetic. This means the pobability density ψ n00 is spheically symmetic. Physically, l = m l = 0 implies that L = 0, i.e., the electon has eo angula momentum. L = 0 is consistent with spheically symmety, since the L vecto picks out a diection in space. Classically, L is fixed and pependicula to the plane of the obit fo cental foce motion. Hence, specifying L immediately tells you the plane of the obit. In quantum mechanics, L cannot be pecisely specified accoding to the uncetainty pinciple, 10
but if it is non-eo, ψ nlml will no longe be spheically symmetic (the obit becomes moe plana). Recall, howeve, that the pobability density ψ nlml is always independent ofφ. As an example of a state with l 0, let us conside a p state, i.e., l = 1. Thee ae thee possible values of m l : 1, 0, +1. Fo m l = 0, Θ(θ) Φ(φ) Θ(θ) cos θ. (4) The figue to the ight shows a pola diagam of cos θ. The distance fom the oigin to the cuve is popotional to the value of cos θ fo any θ. Fo example, fo θ = 0 (along axis), the distance fom the oigin to the cuve is maximum because cos θ is maximum at θ = 0. The thee-dimensional angula θ pobability distibution can be visualied by otating the pola diagam about the -axis though 360, i.e., the full ange of values fo φ (since the pobability density is independent ofφ ). Fo m l = ±1, Θ(θ) sinθ and Φ(φ) e ±iφ. The angula pobability density is given by Θ(θ) sin θ. A pola diagam is shown to the left. Note that fo m l = 0, the pobability that the electon is along the -axis is highest. Fo m l = ± 1, the pobability density is highest in the x-y plane. Consequently the m l = 0 wave function is called the p wave function (note thatψ n10 ) and the m l = ±1 wave functions ae associated with the p x and p y wave functions. 5 [Show plots of angula pobability densities fo vaious values of l and m l. Also, see www.falstad.com fo H-atom quantum-state simulations.] Hydogen-like Atoms Fo a one-electon atom of atomic numbe Z, the potential enegy is U() = kze. (43) The solution to the TISE poceeds in an identical fashion to that fo the H atom (Z = 1). Of couse, the Θ(θ) and Ф(φ ) angula wave functions ae unchanged, but the adial functions ae diffeent because of the diffeent length scale (attaction of the electon to the nucleus is stonge, so the electon will, on aveage, be close to the nucleus than in the case of the H atom). 6 Also, the enegy will be diffeent: E = Z (13.6 ev), (44) n which, as expected, is the same as that obtained fom the Boh model. The enegy simply eflects the stonge binding. 5 The p x and p y wave functions ae eally supepositions of the m l = ±1 states. 6 Show pictue of how length scale deceases fom a B to a B /Z (one-electon wave functions, Eisbeg & Resnick). 11