H atom solution. 1 Introduction 2. 2 Coordinate system 2. 3 Variable separation 4

Transcription

1 H atom solution Contents 1 Introduction 2 2 Coordinate system 2 3 Variable separation 4 4 Wavefunction solutions Solution for Φ Solution for Θ Combined Φ and Θ solution Solution for R Complete wavefunction solution Electron in the H atom Energy of the electron Radial function Angular function Charge distribution Radial distribution Three dimensional charge distribution s orbitals p orbtials d orbitals Quantum numbers Principal quantum number, n Orbital quantum number, l Magnetic quantum number, m Spin quantum number

2 8 Exclusion principle 22 1 Introduction The H atom is an example of applying the Schrödinger equation to solve the energy of the electron in a central potential. The solution can also be used for other one electron systems. It is the only physical system for which a full solution for the wavefunction is possible, excluding spin. The H atom solutions can be extended, with approximations, to atoms with multiple electrons to obtain approximate solutions of the electron energies that match with experimental results. The H atom consists of a nucleus with one proton (unit positive charge, Z = 1) with one electron. The mass of the nucleus is approximately 1840 times greater than that the electron so that it is a valid approximation to consider the nucleus to be stationery. A more accurate model would consider the nucleus stationery but replace the electron mass by the reduced mass give by µ e = m e m n m e + m n (1) where µ e is the reduced mass and m e and m n are the electron and nucleus mass respectively. For the H atom, since m n 1840 m e, equation 1 gives µ e = 1840 m 1841 e or m e. For all practical purposes we can take the mass to be the mass of the free electron (i.e kg). 2 Coordinate system The H atom is an example of a system under a central potential i.e. the potential experienced by an electron is inversely proportional to the distance. Thus, an electron located at a distance r from the nucleus, experiences a potential given by V (r) = Ze2 (2) 4πɛ 0 r where ɛ 0 is the permittivity of free space with a value of F m 1 and Z is the number of positive charges in the nucleus. For the H atom, Z = 1. In a 3D carteisan coordinate system, equation 2 is rewritten as V (x, y, z) = Ze 2 4πɛ 0 x2 + y 2 + z 2 (3) 2

3 Figure 1: The spherical coordinate system, showing r, θ and φ. The potential function, V, is then used in the three dimensional time independent Schrödinger equation 2 2m e 2 ψ + V ψ = E ψ (4) where ψ is the 3D wavefunction of the electron. It is possible to solve this system using Cartesian coordinates and obtain a solution of ψ in terms of (x, y, z). But this solution is complicated since variable separation cannot be done, V (x, y, z) involves all three coordinates. Given that the H atom potential is spherically symmetric, since it only depends on distance, r, it would be easier to solve this system in the spherical coordinate system. In the spherical system, (x, y, z) is replaced by (r, θ, φ), as shown in figure 1. In the Cartesian system the limits for (x, y, z) are ( to ). For the spherical coordinate system the limits are r 0 to θ 0 to π φ 0 to 2π (5) 3

4 The two coordinates are related by x = r sin θ cos φ y = r sin θ sin φ z = r cos θ (6) From equation 6 it is possible to obtain (r, θ, φ) in terms of (x, y, z). This gives r = x 2 + y 2 + z 2 cos θ = tan φ = y x (7) z x2 + y 2 + z 2 The advantage of using spherical coordinate system is that the potential is only a function of r and not θ or φ. Hence it is possible to use variable separation to separate the electron wavefunction into function of r, θ and φ alone. This will help in solving the Schrödinger equation to obtain electron energies and probabilities, i.e. wavefunctions. 3 Variable separation Consider the time independent Schrödinger equations, as shown in equation 4. The potential for the H atom is given by 2 with Z = 1. E is the total energy of the electron in the H atom, which we want to obtain. 2, called the Laplacian or the Laplace operator, is a differential operator that depends on the coordinate system. For the Cartesian system this just becomes 2 = 2 x y z 2 (8) which can be substituted in the Schrödiger equation. But given that the H atom solution is in the spherical coordinate system the Laplacian needs to be converted to this coordinate system. This can be done by using the chain rule for partial derivatives, x = r r x + θ θ x + φ φ x (9) with similar expressions for the y and z terms. Using equation 9 it is possible to write equation 8 in spherical coordinates and the final expression is given 4

5 by 2 = 1 r 2 r (r2 r ) + 1 r 2 sin θ θ (sin θ θ ) + 1 r 2 sin 2 θ 2 φ 2 (10) A detailed derivation is given in Planet math website. An advantage of using the spherical coordinate system is that since the potential only depends on r, see equation 2, it is possible to separate the electron wavefunction into functions of the individual variables. Thus it is possible to write ψ(r, θ, φ) = R(r)Θ(θ)Φ(φ) (11) where R(r), Θ(θ), and Φ(φ) are functions of r, θ, φ alone. This will help in simplifying the Schródinger equation to the individual terms which can be solved to obtain the electron wavefunction. Consider the effect of variable separation, equation 11, on the H atom equation, 4. Substituting the various terms, equation 4 becomes [ 2 1 d 2m e r 2 R dr (r2 ) dr ] dr + 1 d dθ (sin Θ sin θ dθ dθ ) + 1 d 2 Φ Φ sin 2 Ze2 θ dφ 2 4πɛ 0 r = E (12) Rearranging equation 12, by multiplying with 2mer2 sin 2 θ it is possible to 2 write sin 2 θ R d dr (r2 dr dr ) sin θ Θ d dθ (sin θ dθ dθ ) 2Ze2 m e r sin 2 θ 4πɛ 0 2 2Em2 e sin 2 θ 2 = 1 d 2 Φ Φ dφ 2 (13) The left hand side is a function of (r, θ) only while the right hand side is a function of φ only. By the principle of variable separation the two sides must be equal to a constant, independent of (r, θ, φ). It is possible to do a further variable separation to separate the r and the θ part. This can be done by equating the two sides of equation 13 to a constant, m 2. Thus, equation 13 reads sin 2 θ R d dr (r2 dr dr ) sin θ Θ d dθ (sin θ dθ dθ ) 2Ze2 m e r sin 2 θ 4πɛ 0 2 2Em2 e sin 2 θ 2 = m 2 1 Φ d 2 Φ dφ = 2 m2 (14) There is a particular reason for choosing the constant as m 2. Later, this constant m will be called the magnetic quantum number. The top half 5

6 of equation 14 containing the r and θ terms can be further rearranged to separate the two variables. 1 d dr (r2 R dr dr ) 2Ze2 m e r 4πɛ 0 2 2Em2 e 2 = 1 Θ sin θ d dθ (sin θ dθ dθ ) m2 sin 2 θ (15) Once again we have an equation where the left side is only a function of r and the right side only a function of θ. Using the concept of variable separation once again both sides must be equal to a constant, and this constant can be set to l(l + 1). Later we shall call this l the orbital quantum number. Again this particular choice of constant is to make the solution easier to appreciate. Thus, using equations 14 and 15, it is possible to write the general Schrödinger equation, containing ψ(r, θ, φ) into three equations, each containing only one variable. 1 d dr r 2 (r2 dr dr ) + 2m ] e [E + Ze2 2 4πɛ 0 r 2 l(l + 1) R = 0 2m e r 2 ] 1 d dθ (sin θ [l(l sin θ dθ dθ ) + + 1) m2 sin 2 Θ = 0 θ d 2 Φ dφ + 2 m2 Φ = 0 (16) These three equations can be solved separately to obtain the values for (R, Θ, Φ) and combining them together will give the electron wavefunction i.e. ψ. The solution of the Schrödinger equation for the H atom gives the result that the energies of the electrons are quantized i.e. they can only take specific values. 4 Wavefunction solutions 4.1 Solution for Φ Consider the equation for Φ, shown in equation 16. d 2 Φ dφ 2 + m2 Φ = 0 (17) This is a fairly straightforward second order differential equation whose solution can be written in the form Φ = A exp(±imφ) (18) 6

7 A is a normalization constant which can be obtained by setting 2π which gives the normalized solution as 0 Φ Φdφ = 1 (19) Φ = 1 2π exp(imφ) (20) By convention only the positive value of m is considered. This would be useful when plotting the wavefunctions and the probabilities. 4.2 Solution for Θ Consider now the equation for Θ, as shown earlier in equation 16. ] 1 d dθ (sin θ [l(l sin θ dθ dθ ) + + 1) m2 sin 2 Θ = 0 (21) θ This can be rearranged to write as [ 1 sin θ d ] dθ (sin θ Θ dθ dθ ) + sin 2 θ l(l + 1) = m 2 (22) Equation 22 can be simplified by making the substitution µ = cos θ. Hence dµ = sin θdθ. And further d dθ = d dµ dµ dθ = sin θ d dµ (23) Making the substitution, equation 22 becomes ( d (1 µ 2 ) dθ ) ) + (l(l + 1) m2 Θ = 0 (24) dµ dµ 1 µ 2 Solutions to equation 24 are called Associated Legendre equations. These are named after the French mathematician Adrien Marie Legendre ( ) and are essentially recursive polynomial solutions to the differential equation. It is found that the solution depends on both m and l with the constraints that l is a positive integer and m is also an integer that can take values from l to +l i.e. a total of (2l + 1) values. 7

8 4.3 Combined Φ and Θ solution The individual solution to Θ and Φ can be combined into a new function called the spherical harmonics, Yl m (θ, φ). The normalized form of the spherical harmonics function is given by Y m l (θ, φ) = ( 1) m [ 2l + 1 4π ] 1/2 (l m)! exp(imφ) Pl m (cos θ) (25) (l + m)! where Pl m (cos θ) is the associated Legendre polynomials. There are different values for the polynomial depending on l and m. These correspond to the wavefunctions of the different atomic orbitals. The spherical harmonics gives the angular spread of the electron wavefunction i.e. the θ and φ dependence. 4.4 Solution for R Consider the radial part, R(r), of the electron wavefunction given in equation d dr r 2 (r2 dr dr ) + 2m ] e [E + Ze2 2 4πɛ 0 r 2 l(l + 1) R = 0 (26) 2m e r 2 Again this can be rearranged to be written as 2 d dr 2m e r 2 (r2 [V dr dr ) + (r) + l(l + 1) 2 2m e r 2 ] R = ER (27) where V (r) is given by equation 2. This expression can be simplified by making the variable substitution u = rr. Then d dr dr (r2 ) = 2r dr dr dr + r2 d2 R dr 2 = r d2 u dr 2 (28) Making this substitution in equation 27 gives [ ] 2 d 2 u 2m e dr + V (r) + 2 l(l + 1) u = Eu (29) 2 2m e r 2 For the H atom (or H like atom) making the substitution for the potential function, equation 2, equation 29 becomes [ ] 2 d 2 u 2m e dr + Ze2 2 4πɛ 0 r + 2 l(l + 1) u = Eu (30) 2m e r 2 8

9 Equation 30 is similar to a one dimensional Schrödinger equation of wavefunction u where the potential function is given by [ ] V (r) = Ze2 4πɛ 0 r + 2 l(l + 1) for r > 0 2m e r 2 (31) V (r) = for r < 0 By convention an electron in an atom is defined to have total energy E < 0 since this electron must be bound to the atom and cannot escape. The solution to the H atom is to get the quantized values of these energies. Rewriting equation 30 gives d 2 u dr 2 + [ 2m eze 2 4π 2 ɛ 0 r ] l(l + 1) + u = 2m ee u (32) r 2 2 This equation can be solved by suitable substitutions. After normalization the solution depends on two quantum numbers n and l and is given by R nl (r) = ( 2Z na 0 ) 3/2 [ ] 1/2 (n l 1)! exp( ρ 2n(n + l)! 3 where a 0 is a constant and is given by 2 ) ρl L 2l+1 n+l (ρ) (33) a 0 = 4π 2 ɛ 0 m e e 2 = nm (34) This constant is called Bohr radius and represents the most probable distance for finding the electron in the H atom. The dimensionless quantity, ρ, is given by ρ = 2Z na 0 r (35) The quantity L 2l+1 n+l (ρ) is called the Laguerre polymonial and arises because of the recursive nature of the solution to the second order differential equation involving r, equation 30. The quantum numbers n and l involved in the solution to R(r) are called the principal and orbital quantum numbers respectively with the added constraint that l can only take values from 0 to (n 1). This is because (n l 1) must be positive since factorial of a negative number cannot exist, see equation 33. 9

10 Table 1: Electron energies in the H atom n Binding energy (ev ) Spectral series Lymann Balmer Paschen Brackett Complete wavefunction solution The complete solution for the electron wavefunction can be combined by combining the individual solutions for R, Θ, and Φ or R and Y. These solutions are a function of three integers n, l, m which cause quantization of the electron wavefunctions. While it is cumbersome to write the general normalized expression for the electron wavefunction it is possible to write specific wavefunctions for different values of these quantum numbers. Substituting the wavefunction in the H atom equation 4, it is possible to calculate the energy of the electron in the H atom. This energy depends only a single quantum number n and is given by E = m ee 4 Z 2 8ɛ 2 0h 2 n 2 = 13.6Z2 ev (36) n2 The negative sign means that the electron is bound to the nucleus and cannot escape without supplying an external energy. This energy is called the binding energy of the electron and depends on the value of n. 5 Electron in the H atom 5.1 Energy of the electron Consider the energy of the electron in the H atom, given by equation 36, with Z = 1. The values of n, the principal quantum number are 1,2,3... For the first 5 values of n the electron energies are given in table 1. As the value of n increases the energies are closer to 0 i.e. the electron is less under the influence of the nucleus. The binding energy corresponds to the minimum energy required remove an electron from that n value, i.e. to ionize the atom. For an electron transition from a higher value of n to lower value of n the difference in energies is given out as light. These electronic transitions form the characteristic H spectra. Thus transition from n = 2 to n = 1 has an energy difference of 10.2 ev and the wavelength is 122 nm. This lies in the 10

11 R(r)/(Z/a 0 ) 3/ R(r) for 1s r/a 0 Figure 2: Plot of the radial function, R(r), vs r for the 1s H atom orbital. The axes are normalized. Plot generated using MATLAB. UV region and is part of the Lymann series (transitions to n = 1). Similar transitions are shown in table Radial function Consider the radial part of the electron wavefunction, given by R(r), equation 33. For n = 1 the only value of l is 0 and the function is given by R 10 (r) = 2 ( Z a 0 ) 3/2 exp( Zr a 0 ) (37) This corresponds to the 1s orbital, with Z = 1 for H, plotted in figure 2. Equation 37 is an exponential function, as seen from figure 2. For n = 2, there exists two values for l i.e. 0 and 1. These are the 2s (n = 2, l = 0) and 2p (n = 2, l = 1) orbitals. For H atom, these orbitals are R 20 (r) = 2 ( Z 2a 0 ) 3/2 (1 Zr 2a 0 ) exp( Zr 2a 0 ) 2s R 21 (r) = 2 ( Z 2a 0 ) 3/2 Zr exp( Zr ) 2p 2a 0 2a 0 (38) The wavefunctions are plotted in 3. The 2s wavefunction passes through zero at r = 2a 0. This point is called a node. For the 2p there is a node at r = 0 corresponding to the nucleus, but this usually not considered since an electron cannot exist in the nucleus. For n = 3 there are 3 values for l i.e. 0,1, and 2. These correspond to the 11

12 0.8 R(r) for 2s and 2p R(r)/(Z/a 0 ) 3/ s 2p r/a 0 Figure 3: Plot of the radial function, R(r), vs r for the 2s and 2p H atom orbital. The axes are normalized. Plot generated using MATLAB. 0.4 R(r) for 3s, 3p, and 3d R(r)/(Z/a 0 ) 3/ s 3p 3d r/a 0 Figure 4: Plot of the radial function, R(r), vs r for the 3s, 3p, and 3d H atom orbital. The axes are normalized. Plot generated using MATLAB. orbitals 3s, 3p, and 3d respectively. The radial functions are R 30 (r) = ( Z a 0 ) 3/2 (6 6( 2Zr 3a 0 ) + ( 2Zr 3a 0 ) 2 ) exp( Zr 3a 0 ) 3s R 31 (r) = ( Z a 0 ) 3/2 (4 2Zr 3a 0 ) ( 2Zr 3a 0 ) exp( Zr 3a 0 ) 3p R 32 (r) = ( Z a 0 ) 3/2 ( 2Zr 3a 0 ) 2 exp( Zr 3a 0 ) 3d (39) Once again these can be plotted, see figure 4. For the 3s orbital there are 2 nodes, also seen from equation 39 which has a quadratic equation in the expression. For the 3p there is one node. Thus in general R nl (r) will have (n l 1) nodes, excluding the node at r = 0. 12

13 5.3 Angular function The function Y lm, equation 25, gives the angular distribution of the electron wavefunction i.e the distribution in θ and φ. This depends on two quantum numbers l and m with the constraint that m can take integer values from l to l i.e (2l + 1) values. Consider the case when l = 0, the only value of m is also 0 and the angular distribution function for this, equation 25 is Y 0 0 (θ, φ) = 1 4π (40) This corresponds to the s orbital and it is spherically symmetric since there is no dependence on θ and φ. The radial distribution for the s orbitals depends on the value of n, seen from equations 37, 38, 39. Overall the product of both is also spherically symmetric. Consider the case when l = 1, this is the p orbital and it can take 3 values for m, i.e. -1,0,1. The corresponding values of Y ml are Y 1 1 = 6 4π exp( iφ) 1 cos 2 θ 3 Y1 0 = 4π cos θ 3 Y1 1 = 8π exp(iφ) 1 cos 2 θ (41) Similarly for l = 2 there are 5 values for m and hence 5 functions for Yl m. It is possible to look at the angular distribution of the electron probability (i.e. charge distribution). 6 Charge distribution For an electron defined by a wavefunction, ψ, the charge distribution is given by Charge distribution = e ψ ψdv (42) V where ψ is the the complex conjugate of the normalized electron wavefunction, ψ and the integration is over a volume V and dv is the volume element in the appropriate coordinate system used. Since e is a constant the term we are interested is in the integral. For the electron in the H atom since the wavefunction is obtained by converting the system into spherical coordinates the volume element term is given dv = r 2 sin θ dr dθ dφ (43) 13

14 The limits for the coordinates are given by equation 5. For a spherically symmetric wavefunction, like the s orbital (l = 0 and hence m = 0) the volume element, equation 43 simplifies to dv = 4πr 2 dr (44) In the H atom solution, there are two parts to the electron wavefunction, one is the radial part given by R nl (r) and the other the angular distribution given by Yl m (θ, φ). Since the total electron distribution is three dimensional it is easier to consider these two parts separately first before putting together the final wavefunction. 6.1 Radial distribution The radial distribution of the electron is given by Radial distribution = r 2 R (r)r(r) dr = r 2 R 2 (r) dr (45) This is because R(r) is a real expression and hence its complex conjugate is the same as the function itself i.e. R (r) = R(r). It is possible to plot this distribution as a function of r for the different orbitals. Consider the radial function for the 1s orbital given in equation 46. R 10 (r) = 2 ( Z a 0 ) 3/2 exp( Zr a 0 ) (46) The radial distribution function for this, using equation 45, is given by r 2 R 2 10(r) = 4r 2 ( Z a 0 ) 3 exp( 2Zr a 0 ) (47) This function is plotted as a function of r in figure 5. There is a maximum at r = a 0. This corresponds to the most probable location of the 1s H atom. Also, the probability of finding the electron at the nucleus is zero, since there is a r 2 term in equation 47. Hence the solution to the electron wavefunction naturally prevents the electron from falling into the nucleus. It is possible to similarly plot the 2s and 2p orbitals. The radial distribution functions for these, based on the R(r) functions in equation 38 is r 2 R 20 (r) = 4r 2 ( Z 2a 0 ) 3 (1 Zr 2a 0 ) 2 exp( Zr a 0 ) r 2 R 21 (r) = 4r 2 ( Z 2a 0 ) 3 ( Zr 2a 0 ) 2 exp( Zr a 0 ) (48) 14

15 r 2 R 2 (r)*(a 0 /Z) s radial distribution r/a 0 Figure 5: Plot of the radial distribution function, R(r), vs r for the 1s orbital. The axes are normalized. Plot generated using MATLAB. r 2 R 2 (r)*(a 0 /Z) s and 2p radial distribution 2p 2s r/a 0 Figure 6: Plot of the radial distribution function, R(r), vs r for the 2s and 2porbital. The axes are normalized. Plot generated using MATLAB. 15

16 The distribution functions are plotted in figure 6. There is one node for the 2s orbital (at r = 2a 0 ), similar to that seen in figure 3 and the most probable location of the electron is at r = 5a 0. Similar plots could be drawn for higher order orbitals. It is harder to visualize the angular distribution separately. But this can be combined with the radial distribution to give the total three dimensional charge distribution. 6.2 Three dimensional charge distribution The three dimensional charge distribution of the electron in the various orbitals is given by equation 42. Ignoring the e terms the charge distribution is a plot of ψ ψdv, where ψ is the electron wavefunction and dv is the volume element. To understand this charge distribution we can look at charge distribution for different values of the quantum number, l i.e. the angular quantum number s orbitals Consider the electron in the 1s orbital (n = 1,l = 0). This is the ground state of the electron in the H atom and the normalized wavefunction is given by ψ 100 = 1 ( Z ) 3/2 exp( Zr ) (49) π a 0 a 0 Taking the complex conjugate and multiplying the charge distribution is given by ψ ψ dv = 4πr 2 ( 1 π ) ( Z a 0 ) 3 exp( 2Zr a 0 )dr (50) This is a spherical symmetric function and hence the 3D plot should be a sphere. This is seen in figure 7, This is a 3D plot of a constant probability surface represented by equation 50. Similar plots for other s orbitals will all be spheres since there is no radial dependence p orbtials For the electron in the p orbital the value of l = 1. Hence the smallest value of n that is possible is 2 and for each l value there are 3 m values, i.e. -1,0,1. Hence there are 3 p orbitals and are usually denoted p x, p y, and p z. The plot of the 2p orbitals is shown in figure 8. There are 3 2p orbitals and they are oriented along the axes i.e. x, y, and z. They are respectively called p x, 16

17 Figure 7: 3D plot of the 1s charge distribution. The plot was generated using MATLAB and is a plot of a constant probability surface. Figure 8: 3D plot of the 2p charge distribution. The plot was generated using MATLAB and is a plot of a constant probability surface. 17

18 p y, and p z. The corresponding wavefunctions (used to calculate the charge distributions) are ψ 210 = 1 4 2π ( Z ) 3/2 ( Zr ) exp( Zr ) cos θ a 0 a 0 2a 0 ψ 21±1 = 1 8 π ( Z a 0 ) 3/2 ( Zr a 0 ) exp( Zr 2a 0 ) sin θ exp(±iφ) (51) For purpose of plotting the orbitals the exp(±iφ) function is converted into a summation of cos and sin functions so that all three wavefunctions are written as 1 ψ 210 = 4 2π ( Z ) 3/2 ( Zr ) exp( Zr ) cos θ a 0 a 0 2a 0 1 ψ 211 = 4 2π ( Z ) 3/2 ( Zr ) exp( Zr ) sin θ cos φ a 0 a 0 2a 0 ψ 21 1 = 1 8 π ( Z a 0 ) 3/2 ( Zr a 0 ) exp( Zr 2a 0 ) sin θ sin φ (52) The fact that p orbitals have charge (or electron) densities in certain directions is of great importance in bonding. This is responsible for the directional bonding in many of the covalently bonded compounds d orbitals For the d orbitals the value of l = 2. Hence there are 5 d orbitals and they are usually denoted as d z 2, d xy, d yz, d xz, and d x 2 y2. Again the subscripts indicate the predominant directions of the electron distribution in space. The charge distributions are shown in figure 9. A completely filled d or p or f orbital will be spherically symmetric. This can be seen by taking the charge distribution for all the individual orbitals and adding them together. f orbitals have an even more complex charge distribution and the electrons are tightly confined. There are a total of 7 f orbitals (l = 3 and m = 3 to 3). f orbitals rarely participate in bonding due to their confined nature. For certain transition metals the d orbitals can mix with the s orbitals and influence the bonding. 7 Quantum numbers The solution to the one electron atom (H or H like) generates 3 quantum numbers. There is also an additional quantum number that is needed in order to explain the behavior of electrons in a magnetic field (fine structure 18

19 Figure 9: 3D plot of the 3d charge distribution. The plot was generated using MATLAB and is a plot of a constant probability surface. of the spectroscopic lines). Together these four quantum numbers define the electron in the H atom. 7.1 Principal quantum number, n In the one electron system, the principal quantum defines the energy of the electron, by equation 36. With increasing n, the energy reduces since the maximum probability of finding the electron moves away from the nucleus (larger r). 7.2 Orbital quantum number, l For a given value of n the orbital quantum number along with the magnetic quantum number (m) define the spatial distribution of the electron in space. l = 0 and m = 0 corresponds to a spherically symmetric distribution while for all other values the distribution is more complex. The orbital quantum number is present in the expression for the potential experienced by an electron in the H atom, see equation 31. For l = 0 the potential is similar to a normal central potential, while for other values of l the potential is lowered due to shape of the charge distribution, given by l. This represents the centrifugal barrier that keeps the electron away from the nucleus. For l = 0, there is a theoretical possibility of the electron wavefunction being found in the nucleus (since ψ(r) is not zero at r = 0, though the probability term r 2 ψ ψ is still zero) but for other orbitals the l term makes sure that the electron cannot 19

20 be found near the nucleus. The orbital quantum number does not affect the total energy, since this depends only on n for a one electron system. But for a multi electron system, there will be splitting in the energies of the electrons for different values of l, (same n). The orbital quantum number defines the angular momentum of the electron in the H like atom. The angular momentum, L, of the electron is quantized and is given by L = l (l + 1) (53) For l = 0 (s orbital) the angular momentum is zero since the distribution is spherically symmetric. For other values of l there exists a finite value for L. 7.3 Magnetic quantum number, m The magnetic quantum number is related to l and defines the behavior of the electron in an external magnetic field. Experiments on atomic spectroscopy show that in an external magnetic field the emission lines from an atom split. This splitting is related to the quantization of the orbital angular momentum in an external field. If the direction of the field is taken to be z, then the angular momentum in that direction is given by L z = m (54) For the value of m = 0, corresponding to the p z or d z 2 the quantization is zero since the charge distribution is parallel to the external field. For other charge distributions the orbital angular momentum is at angle to the external field and hence the component of L along the z is given by the non-zero value of m. Values of m can be positive or negative since for an external field L z can be positive of negative. For non-zero values of m the net angular momentum L will be at an angle to the z axis and the value of the angle, θ, is given by cos θ = m l(l + 1) (55) Consider a value of l = 2 then the values of m are 2, 1, 0, 1, 2 the corresponding angles are 90 (for m = 0), 66 (for m = 1) and 35 (for m = 2). The quantization of angular momentum is important because it affects the rules governing the interaction of electrons in an atom with light i.e. atomic spectroscopy. When an electron interacts with light of sufficient energy it can get excited from a lower energy level (lower n) to a higher energy level. While energy is conserved in this process angular momentum must also be conserved. Since the photon has a finite non-zero momentum given by, the 20

21 value of l must also change. These are called selection rules and for atomic transition (due to absorption of light) the rules are given by l = ±1 and m = 0, ±1 (56) Thus an electron can be excited from 1s to 2p since according to equation 56 l = 1 but not to 2s since l = 0. Similarly an electron from 2s can be excited to 3p but not to 3s ( l = 0) or 3d ( l = 2). The same rules are also valid for photon emission when the excited electron comes back to the ground state. 7.4 Spin quantum number The orbital quantum number l and magnetic quantum number m come naturally from the solution to the Schrödinger equation for the H atom. These also explain the splitting in atomic spectral lines with and without a magnetic field. A further splitting of the spectral lines, called fine structure, was observed by Stern and Gerlach which could only be explained by considering that the electrons had an intrinsic angular momentum apart from the orbital angular momentum. This intrinsic angular momentum is denoted as Spin, with symbol S. The magnitude of S is given by a new spin quantum number, s and is given by S = s(s + 1) S z = m s (57) s = ± 1 2 m s is called the spin magnetic quantum number (sometimes magnetic quantum number is denoted m l ) and represents the z component of the intrinsic angular momentum. Thus, s can have 2 values ± 1 2. It is possible to combine both L and S and define a total angular momentum, represented by J. Again J can be defined by a quantum number j such that J = L + S J = j(j + 1) J z = m j (58) Like l and s, j is also quantized but it can take fractional values since it is a vector addition of two angular momentum terms. The concept of J becomes important when considering the behavior of a solid in a magnetic field. 21

22 8 Exclusion principle The four quantum numbers n, l, m l and m s define the electron in the H or H like system. For the H atom the quantum number n defines the total energy, per equation 36, while n, l, and m l define the spatial distribution of the wavefunction. For each set of these three values there can only be two values of the spin quantum number m s (i.e. ± 1 ). Thus each electron in a 2 H atom has an unique set of four quantum numbers and no two electrons can have the same set of these quantum numbers. This is called Pauli s exclusion principle. The exclusion principle determines the filling up of electrons in a multi electron system (an atom or a solid) and is the basis for determining the periodic table. Thus for n = 1 there can be a maximum of two electrons, with quantum numbers, (1, 0, 0, 1) and (1, 0, 0, 1 ). The 2 2 corresponding elements are H and He. For n = 2, there are 8 electrons (since there can be s and p orbitals). For n = 3, there are 18 electrons (s, p, and d orbitals). But the 4s level fills up before the 3d levels so that the 3d transition elements are found in the fourth row of the periodic table. This is because for a multi electron system the interaction between the individual electrons also matter and hence equation 36 is only an approximation. For n = 4 there are 32 electrons, since this also includes the f orbitals. This in turn can explain the arrangement of atoms in the periodic table. References The following source materials were used for preparing this handout 1. Principles of modern physics by Robert B. Leighton, McGraw-Hill Book company, Introduction to quantum mechanics by Linus Pauling and E. Bright Wilson, CBS Publishers, Principles of electronic materials and devices by S.O. Kasap, McGraw- Hill India, 2007 The PDF was prepared in L A TEX, using Texmaker and MiKTeX compiler. The plots were generated in MATLAB. The MATLAB code for figures 7, 8, and 9 were adapted from a code by Peter van Alem from MATLAB Central (file name Orbital.m). Thanks to Prof. John H. Weaver, University of Illinois at Urbana-Champaign, for comments and suggestions. 22