Chapter 6 Quantum Theory of the Hydrogen Atom 1
6.1 Schrodinger s Equation for the Hydrogen Atom Symmetry suggests spherical polar coordinates
Fig. 6.1 (a) Spherical polar coordinates. (b) A line of constant zenith angle q on a sphere is a circle whose plane is perpendicular to the z axis. (c) A line of circle constant azimuth angle f is a circle whose plane include the z axis. 3
Cartesian Coordinates Cartesian coordinates. A differential volume in Cartesian coordinates. 4
Cylindrical coordinates 5
Spherical coordinates Spherical coordinates. A differential volume element in spherical coordinates. 6
7 ),, ( r 0 U E m z y x r e U 0 4 1) Cartesian coordinates (x, y, z) Spherical polar coordinates Schrödinger s equation for H-atom (6.1) (6.) Electric potenti al energy
8 x y z y x z z y x r 1 1 tan cos cos sin sin cos sin r z r y r x
9 0 sin 1 sin sin 1 1 ψ U E m ψ θ r θ ψ θ θ θ r r ψ r r r 0 4 sin sin sin sin 0 E r e mr r r r (6.3) (6.4)
10 Laplacian operator 3 3 1 3 3 1 1 1 3 1 3 1 1 u f h h h u u f h h h u u f h h h u h h h f f x f y f x f 1 1 z f f r r f r f r sin 1 sin sin 1 1 f R f R R f R R R (Cartesian coordinates) (Cylindrical coordinates) (Spherical coordinates)
6. Separation of variables A differential equation for each variable 11
Hydrogen-atom Wave function ψ(r,θ,φ) = R(r) Θ(θ) Φ(φ) = R Θ Φ ψ R dr = Θ Φ = Θ Φ r r dr ψ Θ dθ = R Φ = R Φ θ θ dθ ψ Φ dφ = R Θ = R Θ φ φ dφ (6.5) The partial derivatives become full derivatives because R,, and depend on r,, and only. 1
To separate variables, plug = R into Schrödinger's equation and divide by R. The result is sin θ d dr sinθ d dθ r + sinθ R dr dr Θ dθ dθ 1dΦ mr sin θ e + + + E = 0. Φ dφ 4πε0r (6.6) We have separated out the variable! The term 1dΦ Φ dφ is a function of only. Let's put it over on the right hand side of the equation. This gives us 13
sin θ d dr sinθ d dθ r + sinθ R dr dr Θ dθ dθ mr sin θ e 1 d Φ + + E = -. 4πε0r Φ dφ (6.7) This equation has the form f(r,θ) = g(φ) f is a function of r and only, and g is a function of only. How can this be? The RHS has only in it (but no r and ), and the LHS has only r and in it (but no ). Yup, you heard right! And yet you re telling me LHS = RHS. I repeat, how can this be? 14
Only one way! f(r,θ) = a constant, independent or r,θ, and φ = g(φ) Are you telling me everything is just a constant? Absolutely not! It s just that the particular combination of terms on the LHS happens to add up to a constant, which is the same as the constant given by the particular combination of terms on the RHS. sin θ d dr sinθ d dθ r + sinθ R dr dr Θ dθ dθ mr sin θ e 1 d Φ + + E = constant = -. 4πε0r Φ dφ This is really good. We've taken the one nasty equation in r, and separated it into two equations, one in r, and the other in only. Maybe we can separate the r part? 15
sin θ d dr sinθ d dθ r + sinθ R dr dr Θ dθ dθ mr sin θ e 1 d Φ + + E = constant = -. 4πε0r Φ dφ (6.7) It turns out (although we won't do the math in this course) that the constant must be the square of an integer. If not, our differential equations have no solution. Thus, we can write the RHS of this equation as 1 d d m l (6.8) Where did this m l come from? It s an integer. We just happened to give it that name. 16
The LHS of our big Schrödinger equation also must equal m l. If we set the LHS equal to m l, divide by sin, and rearrange, we get 1 d dr mr e m 1 d dθ r + + E = - sinθ. R dr dr 4πε0r sin θ Θ sinθ dθ dθ (6.9) = constant = l(l+1) m l sin 1 d d sin sin d d l( l 1) (6.10) 1 R d dr r dr dr mr e 4 r 0 E l( l 1) (6.11) 17
Equations (6.8), (6.10), and 96.11) are usually written Equation for Equation for Equation for R d m l d m1 d dθ +1- sinθ + sinθ = 0 sinθ dθ dθ +10 1 d θ dr m e r + + E R = 0 r dr dr 4πε0r r (6.1) (6.13) (6.14) 18
6.3 Quantum Numbers Three dimensions, three quantum numbers 19
We find the first quantum number by solving the differential equation for. d Φ + m Φ = 0 dφ That equation should look familiar to you; you've seen it a number of times before. It has solutions which are sines and cosines, or complex exponentials. We write the general solution Φφ = A e j m φ. We will get the constant A by normalization. (6.15) Now, because and + represent a single point in space, we must have j mφ j m φ + A e = A e π. This happens only for m l = 0, 1,, 3,... 0
Fig. 6. The angle and + both indentify the same meridian plane 1
l is another quantum number, called the orbital quantum number, and the requirement on l can be restated as m l = 0, 1,, 3,..., l. I ll summarize OSE s in a bit. Finally, the radial differential equation is +1 1 d dr m e r + + E R = 0 r dr dr 4πε0r r It can be solved only for energies E which satisfy the same condition as we found on the energies for the Bohr atom: E E = - =, n = 1,, 3... 4 me 1 1 n 3π ε0 n n n is called the principal quantum number. (6.16)
Here s the differential equation for r again: can express the requirement that n=1,,3, and n(l+1) 1We as a condition on + 1 d dr m e r + + E R = 0 r dr dr 4πε0r r Note that the product l(l+1) shows up in the equation for R, and n comes out of solving this equation. Another math requirement for valid solutions is that n(l+1). l: Summarizing our quantum numbers: l = 0, 1,,..., (n-1). n = 1,, 3,... l = 0, 1,,..., (n-1) m l = 0, 1,, 3,..., l (6.17) 3
Rnl lm l m l 4
5 Ex. 6.1 Can you get and E 1 from the ground state wave function for hydrogen?? Table 6.1 put E=E 1, l=0 in Eq. (6.14) a 0 0 3 0 a r e a R 0 1 4 4 0 5 0 3 0 0 3 0 1 7 a r e r a a me a me a 0 0 4 me a 0 4 0 1 3 me ma E
6.4 Principal Quantum Number Quantization of energy 6
E n E 1 (n 1,, 3, n ) How about solar system (e.g. earth and sun) or planetary motion? Is it quantized? Yes, but the separation of permitted energy levels too small to be detected. classical physics planetary motion, not atomic motion!! 7
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6.5 Orbital Quantum Number Quantization of angular-momentum magnitude 9
30 Radial part of Schrödinger s Eq. radial part should be zero 0 1) ( 4 1 0 R r l l E r e m dr dr r dr d r KE orbital KE radial U KE KE E r e U orbital radial 0 4 0 1) ( 1 R mr l l KE KE m dr dr r dr d r orbital radial ) v ( v 1 mr L mr r m m KE orbital orbital (6.14) (6.) (6.19) (6.0)
L mr l( l 1) mr L l( l 1) Electron angular momentum L l( l 1) (6.1) l for =, for earth L (3) l very large.610 34 J s L.710 40 J s l Designation of angular momentum states = 0 1 3 4 5 6 sp d f g h i (sharp) (principal)(diffused)(fundamental):depending on spectrum 31
6.6 Magnetic Quantum Number Quantization of angular-momentum direction 3
Question: What possible significance can a direction in space have for a hydrogen atom? Answer: The answer becomes clear when we reflect that an electron revolving about a nucleus is a minute current loop and has a magnetic field like that of a magnetic dipole. Hence an atomic electron that possesses angular momentum interacts with an external magnetic field B. The magnetic quantum number m l specifies the direction of L by determining the component of L in the field direction. This phenomenon is often referred to as space quantization. 33
If there is a magnetic field in Z-axis, L r mv L Z angular momentum orientation is quantized m l 1 m l L m l m l m l l l 6 0 1 1 l space quantization L Z L Z m l m l 0, 1, l (magnetic quantum number) m l L? (6.) 34
Right-hand rule: When the fingers of the right hand point in the direction of the motion, the thumb is in the direction of L. Fingers of right hand in direction of rotational motion Fig. 6.3 The right-hand rule for angular momentum 35
Fig. 6.4 Space quantization of orbit angular momentum. Here the orbital quantum number is l= and th ere are accordingly l+1=5 possible values of the magnetic quantum number ml, with each v alue corresponding to a different orientation re lative to the z axis. 36
Fig. 6.5 The uncertainty principle prohibits the angular momentum vector L from having a definite direction in space. 37
Uncertainty principle and space quantization Z L i) L Z L Z 0 r v p L Not right! Z m l L L Z ii) l( l 1) L Z Z L 0 OK within uncertainty principle 38
Fig. 6.6 The angular momentum vector L precesses constantly about the z axis. 39
6.7 Electron Probability Density No define orbits 40
In Bohr theory, angular momentum only one value (not considering its orientation) p r,, has a meaning in quantum mechanics. The probability of density to find an electron: R (6.3) Ae im l is symmetrical for Radial part of probability P r dr dv r R dr 0 sin r ddrdr sin d 0 d r R dr (6.5) 41
Figure 6.7 The Bohr model of the hydrogen atom in a spherical polar coordinate system. 4
Radial part Figure 6.8 The variation with distance from the nucleus of the radial part of the electron wave function in hydrogen for various quantum states. The quantity a 4 me = 0.053 nm is the radius of the first Bohr orbit. 0 0 / 43
Probability of Finding the Electron The probability density of the electron at the point r,, is proportional to, But the actual probability of finding it in the infinitesimal volume element dv there is dv. In spherical polar coordinates (Fig. 6.9) Volume element dv ( dr)( rd )( r sind) r sindrd (6.4) Figure 6.9 Volume element dv in spherical polar coordinates. 44
Figure 6.10 The probability of finding the electron in a hydrogen atom in the spherical shell between r and r+dr from the nucleus is P(r)dr. 45
Not symmetric Probability a 0 Figure 6.11 The probability of finding the electron in a hydrogen atom at a distance between r and r+dr from the nucleus for the quantum states of Fig. 6.8. 46
Equation (6.5) is plotted in Fig. 6.11 for the same states whose radial functions R were shown in Fig. 6.8. The curves are quite different as a rule. We note immediately that P is not a maximum at the nucleus for s states, as R itself is, but has its maximum a definite distance from it. (1) The most probable value of r for a 1s electron turns out to be exactly a 0, the orbital radius of a ground-state electron in the Bohr model. (1) However, the average value of r for a 1s electron is 1.5a 0, which is puzzling at first sight because the energy levels are the same in both the quantum-mechanical and Bohr atomic models. This apparent discrepancy is removed when we recall that the electron energy depends upon 1/r rather than upon r directly, and the average value of 1/r for a 1s electron is exactly 1/a 0. 47
Hydrogen 1s Radial Probability 48
Hydrogen s Radial Probability 49
Hydrogen 3s Radial Probability 50
Hydrogen 3p Radial Probability 51
Hydrogen 3d Radial Probability 5
6.8 Radiative Transitions What happens when an electron goes from one state to another 53
The time-dependent wave function n of an electron in a state of quantum number n and energy En is the product of a time-independent wave function n and a timevarying function whose frequency is n e n E n i t * *, n n e E i n t (6.6) 1) electron in a state x x * n n dx x * n dx n does not radiate (6.7) ) electron transition from m n a b * * a b a : : a * a b n * b b probability in n-state probability in m-state 1 m 54 (6.8)
55 real part of (1)+() during transition : electron oscillation with frequency of radiation energy : dx x b dx e x b a dx e x a b dx x a dx b b a a b a x dx b a b a x x m m t E E i m n t E E i n m n n m m m n n m n n m n m n m n n m * 1 * * 1 * * * * * * * * * * * * * (1) () h E E t t h E E t E E n m n m n m where cos cos cos h (6.9) (6.30) (6.30) (6.33)
When the electron is in state n or state m the expectation value of the electron s position is constant. When the electron is undergoing a transition between these states, its position oscillates with the frequency. Such an electron, of course, is like an electric dipole and radiates electromagnetic waves of the same frequency. This result is the same as that postulated by Bohr and verified by experiment. As we have seen, quantum mechanics gives Eq. (6.33) without the need for any special assumptions. 56
6.9 Selection Rules Some transitions are more likely to occur than others 57
The general condition necessary for an atom in an excited state to radiate is that the integral x n * m dx or - * n x m dx (6.34) Exception value <x> of the position of such an electron not be zero, since the intensity of the radiation is proportional to it. Transitions for which this integral is finite are called allowed transitions, while those for which it is zero are called forbidden transitions. Allowed transitions u n *, l, m n', l', m ' dv l l 0 (6.35) Selection rules l m l 1 0, 1 (6.36) (6.37) 58
The change in total quantum number n is not restricted. Equations (6.36) and (6.37) are known as the selection rules for allowed trans itions (Fig. 6.13). 59
Figure 6.13 Energy-level diagram for hydrogen showing transitions all l 1 owed by the selection rule. In the diagram the vertical axis represents excitation energy above the ground state. 60
6.10 Zeeman effect How atoms interact with a magnetic field 61
Zeeman Effect First reported by Zeeman in 1896. Interpreted by Lorentz. Interaction between atoms and field can be classified into two regimes: Weak fields: Zeeman effect, either normal or anomalous. Strong fields: Paschen-Back effect. Normal Zeeman effect agrees with the classical theory of Lorentz. Anomalous effect depends on electron spin, and is purely quantum mechanical. 6
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If you shoot an electron through a region of space with no magnetic field, the electron will experience no deflection (assuming no gravitational forces). - If you shoot an electron through a region of space with a nonzero magnetic field, you know that the electron will experience a deflection. - 65
In an external magnetic field B, a magnetic dipole has an amount of potential energy U m that depends upon both the magnitude of its magnetic moment and the orientation of this moment with respect to the field (Figure 6.15). The torque on a magnetic dipole in a magnetic field of flux density B is B B sin B U m 0 d B 0 sin d Figure 6.15 A magnetic dipole of moment at the angle relative to a magnetic field B. U m 0 cos B U :minimum B B // m (6.38) 66
The magnetic moment of a current loop has the magnitude = I A where I is the current and A the area it encloses. The charge circulating a loop constitutes a current of magnitude I e T ef where T is the orbital period of the electron 67
= IA e L m Figure 6.16 (a) Magnetic moment of a current loop enclosing area A. (b) Magnetic moment of an orbiting electron of angular momentum I. 68
1)magnetic moment, of a current loop I A efr ) electron magnetic moments B e I L mvr efr mfr e m L (6.39) gyromagnetic ratio U m cos e LB m m l l cos l 1, L l l 1 (6.40) U m m l e B m Bohr magneton: (6.41) B 5.788 10 5 ev/t 69
70 depends upon energy depends on n and degenerate by l energy depends on n and splitting Zeeman effect m U l m : 0 : 0 B B m l h E E n m 0 1 l l B m e h B B m e h B B B 4 4 0 0 3 0 0 0 1 (6.43)
1 No magnetic field Magnetic field present Spectrum without magnetic field Spectrum with magnetic field present Figure 6.17 In the normal Zeeman effect a special line of frequency 0 is split into three components when the radiating atoms are in a magnetic field B. One component is 0 and the others are less than and greater than 0 by eb/4m. There are only 71 three components because of the selection rule m l = 0, ±1.
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