PHYS 262. George Mason University. Professor Paul So

Transcription

1 PHYS 6 George Mason University Professor Paul So

2 Chapter 40/41: Quantum Mechanics Wave Functions & 1D Schrodinger Eq Particle in a Box Wave function Energy levels Potential Wells/Barriers & Tunneling The Harmonic Oscillator The H-atom

3 Wave Equation for a Mechanical String y x, t For a wave on a string (1D) moving with speed v, a wave function must satisfy the wave equation (Ch. 15): y x, t, 1 y xt, y xt x v t It has the following sinusoidal form as its fundamental solution:, cos sin y xt A kx t B kx t where k is the wave number and f is the angular frequency of the wave. [A and B determines the amplitude and phase of the wave.]

4 Wave Equation for a String By substituting the fundamental wave function into the PDE, we can arrive at the algebraic relation (dispersion relation) that and k must satisfy: y x, t x k v or vk check Each spatial derivative of y xt, will pull out one k: Acos kx t Bsin kx t A sin kx t B cos x x k k k t So, the nd order spatial derivative gives, y x, t x k sin Ak kx t B kx t cos (Obviously, don t forget the signs.)

5 Wave Equation for a String check.... Each time derivative of y xt, will pull out one : y x, t t Acos kx t B sin kx t A sin kx t B cos kx t t So, the nd order time derivative gives, y x, t t sin A kx t B kx t cos Putting these back into the wave equation, we then have, (Again, don t forget the signs.), 1 yx, t y x t x v t k sin cos Ak kx t B kx t 1 cos v sin A kx t B kx t k v

6 Wave Equation for a String Putting the definitions for and k back into the dispersion relation, we have the familiar relation for wavelength, frequency, and wave speed. vk f v Thus, the fundamental property of a mechanical wave y xt, 1 y xt, is intimately linked to the form of the wave equation! x v t v f v f or k Now, we will try to use the same argument to find a wave equation for a quantum wave function. v

7 Wave Equation for a Quantum Free Particle A free particle has no force acting on it. Equivalently, the potential energy U( x) must be a constant for all x, i.e., F du x dx or U(x) is a constant. x 0 Since the reference point for U( x) is arbitrary, we can simply take U( x) 0. Then, the total energy E of a free particle will simply be its kinetic energy, E 1 mv p m m mv (non-relativistic) Now, from the de Broglie relations, the energy and momentum of this quantum free particle can be related to its wave number k and angular frequency through: h p E hf f E m k h h p k m

8 Wave Equation for a Quantum Free Particle Thus, a correct quantum wave function for this free particle must satisfy this quantum dispersion relation for k and : k * (non-relativistic) m We now assume the same fundamental sinusoidal form for the wave function of a quantum free particle with mass m, momentum p kand energy E : x, t Acoskx t B sinkx t Recall from our discussion on the mechanical wave, we have the following: x take out an overall k factor from x, t t take out an overall factor from x, t

9 Wave Equation for a Quantum Free Particle So, from the quantum dispersion relation, k m We can deduce that the PDE for the quantum wave function for this free particle must involves: x t Putting in other constants so that units are consistent and one additional dimensionless fitting constant C, we have, x, t x, t C m x t

10 Wave Equation for a Quantum Free Particle Now, we substitute our trial quantum wave function x, t Acoskx t B sinkx t into the proposed wave equation to solve for the fitting constant C: xt, cos m x m sin Ak kx t Bk kx t k Acos kx t B kx t m sin xt, C CA kx t B kx t t sin cos sin CB cos kx t CA kx t

11 Wave Equation for a Quantum Free Particle k Equating the two terms and using the equality, we have, m k m sin Acos kx t B kx t CB cos kx t CAsin kx t A CB B CA Substituting the first eq into the second, we have, In order for this equality to be true for all xt,, all coeff s for cos and sin must equal to each other, BC CB C 1 Thus, the fitting constant is C i where i 1.

12 Wave Equation for a Quantum Free Particle Then, finally, putting everything together, we have the desired wave equation for a quantum free particle, x, t x, t i m x t This is the 1D Schrodinger s Equation for a free particle. With B CAiA, the free particle quantum wave function can also be written in a compact exponential form using the Euler s formula, x, t Acoskx t isinkx t xt, Ae ikxt (quantum wave function for a free particle)

13 Free Particle Wave Function & Uncertainty Principle The wave function for a free particle is a complex function with sinusoidal real and imaginary parts A quantum free particle exists in all space,, x & t (wave function extends into all space & time) but p0 & E 0 (energy and momentum is fixed) Note: x p & t E can still be satisfied.

14 More Realistic Particle (Wave Packets) Under more practical circumstance, a particle will have a relatively well defined position and momentum so that both x and p will be finite with limited spatial extents. A more localized quantum particle can not be a pure sine wave and it must be described by a wave packet with a combination of many sine waves. (, ) ( ) ikx xt Ake t dk (a linear combination of many sine waves.) The coefficient A(k) gives the relative proportion of the various sine waves with diff. k (wave number).

15 Wave Packets Recall: Combination of two sine waves more localized than a pure sine wave.

16 Wave Packets (characteristic) p smaller x bigger

17 Wave Packets (characteristic) p bigger x smaller The is consistent with: x p!

18 Quantum Wave Function In QM, the matter wave postulated by de Broglie is described by a complexvalued wavefunction (x,t) which is the fundamental descriptor for a quantum particle. 1. Its absolute value squared ( xt, ) dx gives the probability of finding the particle in an infinitesimal volume dx at time t.. For any Q problem: The goal is to find ( xt, ) for the particle for all time. Physical interactions involves operations (O) on this wave function: O( x, t) Experimental measurements will involve the products, ( xt, ) O( x, t) Re/Im (x,t) x,t (x,t) is a complex-valued function of space and time.

19 The 1D Schrodinger Equation As we have see, ( x, t) ( x, t) U( x) ( x, t) i m x t KE + PE = Total E - the first term ( nd order spatial derivative term) in the Schrodinger equation is associated with the Kinetic Energy of the particle - the last term (the 1 st order time derivative term) is associated with the total energy of the particle - together with the Potential Energy term U(x)(x) the Schrodinger equation is basically a statement on the conservation of energy.

20 The Schrodinger Equation In Classical Mechanics, we have the Newton s equation which describes the trajectory x(t) of a particle: F mx In EM, we have the wave equation for the propagation of the E, B fields: EB, 1 EB, x c t (derived from Maxwell s eqns) In QM, Schrodinger equation prescribes the evolution of the wavefunction for a particle in time t and space x under the influence of a potential energy U(x), U(x) ( x, t) ( x, t) U( x) ( x, t) i m x t (general 1D Schrödinger equation)

21 Wave Function and Probability * ( xt, ) ( xt, ) ( xt, ) is the probability distribution function for the quantum particle. In other words, ( x, t) (shaded area) is the probability in finding the particle in the interval [ xx, dx] at time t. dx Since p( xdx ) ( xt, ) dxis a probability, it has to be normalized! p( x) dx ( x, t) dx 1 (At any instance of time t, the particle must be somewhere in space!)

22 Stationary States For most problems, we can factor out the time dependence by assuming the following harmonic form for the time dependence, i t ( xt, ) ( xe ) ikxt ikx it (Recall the free particle case: ( x, t) Ae Ae e.) With E /, we can rewrite the time exponent in terms of E, ( xt, ) ( xe ) iet / ( xt, ) is a state with a definite energy E and is called a stationary state. ( x) is called the time-independent wave function.

23 The Time-Independent Schrödinger Equation Substituting this factorization into the general time-dependent Schrodinger Eq, we have RHS and, LHS ( xt, ) ie i i ( x) e i ( x) e E ( x) e t t ( xt, ) d ( x) iet/ e x dx iet / iet / iet / d ( x) iet/ e iet / iet/ U( x ) ( x) e E ( x) e m dx d ( x) U( x) ( x) E( x) m dx (time dependence can be cancelled out!) (time-independent Schrodinger equation)

24 More on (time-independent) Wavefunction Note that, ( xt, ) * * iet/ iet/ ( xt, ) ( xt, ) ( xe ) ( x) e ( x) ( x) e ( x) ( x) ( x) * i( Et/ Et/ ) * So, in general, the probability in finding the particle in the interval [a,b] is given by: b pab ( x) dx a p(x) Note: ( x) is not the probability density ( x) is the probability density. a b x

25 More on (time-independent) Wavefunction Other physical observables can be obtained from (x) by the following operation: example (position x): x xp( x) dx x ( x) dx - x is called the expectation value (of x): it is the experimental value that one should expect to measure in real experiments! In general, any experimental observable (position, momentum, energy, etc.) O(x) will have an expectation value given by: O O( x) ( x) dx O can be x, p, E, etc. Note: Expectation values of physically measurable functions are the only experimentally accessible quantities in QM. Wavefunction ( x) itself is not a physically measureable quantity.

26 Solving QM Problems with (timeindependent) Schrodinger Equation Given: A particle is moving under the influence of a potential U(x). Examples: Free particle: U(x) = 0 Particle in a box: U( x) 0, 0 x L, elsewhere Barrier: U( x) U, 0 0 x L 0, elsewhere HMO: 1 U( x) k' x

27 Solving QM Problems with (timeindependent) Schrodinger Equation Solve time-independent Schrodinger equation for (x) as a function of energy E, with the restrictions: d ( x) ( x) and are continuous everywhere for smooth U(x). dx ( x) is normalized, i.e., ( x) dx 1 Bounded solution: ( x) 0 as x Then, expectation values of physical measurable quantities can be calculated.

28 Particle in a Box Classical Picture A 1-D box with hard walls: U(0) U( L) (non-penetrable) A free particle inside the box: U( x) 0 (inside box) No forces acting on the particle except at hard walls. P (in x) is conserved between bounces P is fixed but P switches sign between bounces.

29 Particle in a Box (Quantum Picture) The situation can be described by the following potential energy U(x): U( x) 0, 0 x L, elsewhere The time-independent Schrodinger equation is: d ( x) U( x) ( x) E( x) m dx Recall, this is basically KE + PE = Total E Problem statement: For this U(x), what are the possible wave functions (x) and their corresponding allowed energies E?

30 Wave functions for a Particle in a Box Inside the box, 0 x L, U(x) = 0, and the particle is free. From before, we know that the wave function for a free particle has the following form: ikx ikx (linear combination of the two inside ( x) Ae 1 Ae possible solutions.) p k where A 1 and A are constants that will be determined later. E m m Outside the box, U( x), and the particle cannot exist outside the box and ( x) 0 (outside the box) outside At the boundary, x = 0 and x = L,the wavefunction has to be continuous: inside (0, L) (0, L) 0 outside

31 Wavefunctions for a Particle in a Box Let see how this boundary condition imposes restrictions on the two constants, A 1 and A, for the wave function. Using the Euler s formula, we can rewrite the interior wave function in terms of sine and cosine: ikx e 1 A A cos kx ia A sin kx inside( x) A cos kx i sin kx A cos kx i sin kx 1 1 Imposing the boundary condition at x = 0, (0) cos0 sin 0 A A inside A A i A A 1 1 ikx e 1 0 A A 1 (where C=iA 1 ) inside( x) ia1 sinkx Csinkx

32 Wavefunctions for a Particle in a Box Now, consider the boundary condition at x = L: ( L inside ) C sin kl 0 For a non-trivial solution ( C0), only certain sine waves with a particular choice of wave numbers (k) can satisfy this condition: n kl n n or kn, n1,,3, L This implies that the wavelengths within the box is quantized! n k n L L, n 1,,3, n n Allowed wavefunctions must have wavenlengthes exactly fit within the box!

33 Wavefunction for a Particle in a Box Rewriting this, we have, L n, n 1,,3, n Graphically, it looks like Since k n is quantized, only a discrete set of ( x n ) is allowed as solutions, n n( x) CsinknxCsin x, L n 1,,3, 5 / 3 / / (similar to standing waves on a cramped string)

34 Quantized Energies for a Particle in a Box Since the wave number k n is quantized, the energy for the particle in the box is also quantized: E n kn n n n h or, n1,,3, m m L ml 8mL (n is called the quantum number) Note: the lowest energy is not zero: h E1 0 8mL n = 0 gives (x) = 0 and it means no particle.

35 Probability and Wavefunction Recall that (x) (and not the wavefunction itself (x)) is the probability density function. In particular, ( ) sin n x x dx C dx L gives the probability in finding the particle in an interval [x, x+dx] within the box.

36 Probability in Finding the Particle Notes: The positions for the particles are probabilistic. We just know that it has to be in the box but the exact location within the box is uncertain. Not all positions between x = 0 and L are equally likely. In CM, all positions are equally likely for the particle in the box. There are positions where the particle has zero probability to be found.

37 Probability and Normalization Although we don t know exactly where the particle might be inside the box, we know that it has to be in the box. This means that, ( x) dx 1 (normalization condition) L n L C sin xdxc 1 L 0 So, the normalization condition fixes the final free constant C in the wavefunction, C L. This then gives, ( ) sin n x n x (particle in a box) L L

38 Time Dependence Note that with (x) found, we can write down the full wavefunction for the time-dependent Schrodinger equation as: ( xt, ) ( xe ) iet n n / recall E hf n x n( xt, ) sin e L L iet / / Note that the absolute value for e iet is unity, i.e., iet / iet / iet / 0 e e e e 1 so that n (x,t) = (x) is independent of time and probability density in finding the particle in the box is also independent of time.

39 Finite Square-Well Potential In Newton s mechanics, if E < U 0, a particle will be trapped inside the well. In QM, such a trapped state is called a bound state. If E > U 0, then the particle is not bound. Square-well with finite height For the infinitely deep well (as in the particle in a box problem), all states are bound states. U( x) U elsewhere 0, 0 x L 0, For a finite square-well, there will typically be only a finite number of bound states.

40 Finite Square-Well Potential Similar to the particle in a box problem, U(x) = 0 inside the well, we have or, d ( x) me k ( x), where k dx ( x) Ae Ae ikx 1 ikx inside( x) AcoskxBsinkx and A, B are constants to be determined by boundary conditions and normalization. (inside the well) But for a finite square-well potential, the wavefunction is not identically zero outside the well. The Schrodinger equation is given by: d ( x) dx m mu 0 E U 0 E ( x) ( x), where (outside the well)

41 Finite Square-Well Potential Since U 0 > E, is real and the wavefunction outside the well is given in terms of exponentials instead of harmonic functions: x ( x) Ce De x where C and D are constants to be determined by B.C. and normalization again. For this problem, there is a new type of B.C. at large distances from the origin: wavefunction must remain finite (not blowing up) at large x x ( ) x and ( ) 0 x Ce xl x De x

42 Finite Square-Well Potential For a physical quantum particle, both ( x) and d ( x)/ dx must be continuous at x = 0 and x = L. Matching x0 ( x), inside( x),and xl( x) at x = 0 and x = L will enforce a certain set of allowed functions to be fitted within the well and the bound state energy is correspondingly quantized.

43 Example: e in a Square-Well/Quantum Dot An electron trapped in a Square-Well potential with width L = 0.5nm (~size of an atom) a) What is ground state energy E if this well is infinitely deep U 0 = instead? E, Js 31 9 ml kg m,1 J ev b) Now, back to a finite well with U 6E 9.0eV 0,1 The energy levels for the finite well are given as shown on the next slide. (not derived here)

44 Example: e in a Square-Well/Quantum Dot = 7.6eV = 3.6eV = 0.94eV What is the wavelength of light released if the electron was originally at the 1 st excited state (n=) and relaxed back to the ground state (n=1)? hc hf E E1 hc 140eV s 460nm E E 3.6eV 0.94eV 1

45 Example: e in a Square-Well/Quantum Dot Application: Quantum dots are nanometer-sized particles of a semi-conductor (such as cadmium selenide or gallium arsenide). An electron within a quantum dot behaves much like a particle in a finite square well potential. When a quantum dot is illuminated by a ultraviolet light, the electron within the quantum dot can be excited to a higher energy state (let say, n=3) from ground state (n=1). When it relaxed back to the ground state thru the intermediate state (n=): [3 and 1] photons with lower energy (longer wavelengths in the visible range) can be observed (fluorescence)! D Qdot

46 Tunneling Through a Barrier Consider the following potential barrier: U( x) U, 0 0 x L 0, elsewhere A quantum particle with mass m and energy E is traveling from the left to the right. Classical Expectation (with E < U 0 ): In the region x < 0, the particle is free but when it reaches x = 0, the particle will hit a wall since its E is less than the potential at x =0. It will be reflected back and it could not penetrate the barrier!

47 Tunneling Through a Barrier Quantum Picture: x < 0 and x > L (free space): 0 x L(inside the barrier): The wavefunction for a free particle with definite E and P is sinusoidal, e ikx or e -ikx. E U 0 wavefunction is a decaying exponential e -x. exponential function within barrier

48 Tunneling Through a Barrier If energy is sufficiently high (but still E U 0 ) and the barrier is not too wide so that the exponential decay does not significantly diminish the amplitude of the incidence wave, then there is a non-zero probability that a quantum particle might penetrate the barrier. (reduced amplitude reduced probability but not zero probability!) The transmission probability T can be solved from the Schrodinger equation by enforcing the boundary conditions: mu 0 E L E E T Ge,, G 16 1 U0 U0 (for E/U 0 small)

49 Quantum Tunneling A. I. Kolesnikov et al., Phys. Rev. Lett. (016) A ring-shaped new molecular state for water when a single water molecule is confined inside a hexagonally shaped channel of the gemstone beryl. The H atom delocalizes (spread out) by tunneling to other five classically inaccessible orientational states.

50 Application of Tunneling (STM) Scanning Tunneling Microscope (STM): The tunneling current detected will vary sensitively on the separation L of the surface gap and these variations can be used to map surface features. In a STM, an extremely sharp conducting needle is brought very close to a surface that one wants to image. When the needle is at a positive potential with respect to the surface, electrons from the surface can tunnel through the surface-potential-energy barrier.

51 The Harmonic Oscillator Classically, the harmonic oscillator can be envisioned as a mass m acted on by a conservative force: F k' x (Hooke s Law: mass on a spring). Its associated potential energy is the familiar: 1 U( x) k' x where k is the spring constant. For a classical particle with energy E, the particle will oscillate sinusoidally about x = 0 with an amplitude A and angular frequency. k' m

52 The Harmonic Oscillator The Harmonic Oscillator is important since it is a good approximation for ANY potential U( x) near the bottom of the well.

53 The Harmonic Oscillator For the Quantum analysis, we will use the same form of the potential energy for a quantum Harmonic Oscillator. d m dx ( x) 1 k ' x ( x ) E ( x ) And we have the following quantized energies: d ( x) m1 k ' x E ( x ) The solutions for this ordinary differential equation with the boundary condition ( x) 0 as x are called the Hermite functions: or Boundary condition consideration: U(x) increases without bound as x so that the wavefunction for particle with a given energy E must vanish at large x. ( x) Ce mk ' x dx 1 En n, n0,1,, (ground state n=0)

54 The Harmonic Oscillator Hermite Functions 1 En n, n0,1,, note: wavefunction penetration into classically forbidden regions. note: similar to previous examples, the lowest E state is not zero.

55 The Harmonic Oscillator Probability Distribution Function: Classically, the particle with energy E will slow down as it climbs up on both side of the potential hills and it will spend most of its time near. x A The blue curve depicts this classical behavior and the QM ~ CM as the quantum number n increases.

56 The H-atom In the Schrodinger equation, we have explicitly included the Coulomb potential energy term under which the electron interacts with the nucleus at the origin: 1 e Ur (), 4 r 0 r x y z is the radius in spherical coordinates.

57 The H-atom e - does not exist in well-defined circular orbits around the nucleus as in the Bohr s model. e - in a H-atom should be envisioned as a cloud or probability distribution function. The size and shape of this cloud is described by the wavefunction for the H- atom and it can be explicitly calculated from the Schrodinger equation: 1 e m x y z 4 0 r E (in 3D)

58 Electron Probability Distributions In 3D, the probability in finding the electron in a given volume dv is given by, ( xyz,, ) dv A good way to visualize this 3D probability distribution is to consider a thin spherical shell with radius r and thickness dr as our choice for dv: dv 4 r dr dr r We denote the probability of finding the electron within this thin radial shell as the radial probability distribution function P(r) with: P() rdr dv 4rdr

59 Electron Probability Distributions Examples of the 3-D probability distribution function (electron cloud): The corresponding radial probability distribution function P(r): 4 0 a me 11 m is the Bohr s radius which we have seen previously.

60 More Electron Probability Distributions

61 Quantum Number Recall that for a particle in a 1D box, we have one quantum number for the total energy of the particle. 5 / It arises from fitting the wavefunction [sin (nx/l)] within a box of length L (quantization). 3 / / In the H-atom case, we are in 3D, the fitting of the wavefunction in space will result in additional quantum numbers (a total of 3).

62 Quantum Numbers 1. n Principle Quantum Number: related to the quantization of the main energy levels in the H-atom (as in the Bohr s model). E n 13.6eV, n1,,3, n The other two related to the quantization of the orbital angular momentum of the electron. Only certain discrete values of the magnitude and the component of the orbital angular momentum are permitted:

63 Quantum Numbers. l Orbital Quantum Number: related to the quantization of the magnitude of the e - s orbital angular momentum L. 1, 0,1,,, 1 L l l l n (note: in Bohr s model, each energy level (n) corresponds to a single value of angular momentum. In the correct QM description, for each energy level (n), there are n possible values for L.) 3. m l Magnetic Quantum Number: related to the quantization of the direction of the e - s orbital angular momentum vector. L m, m 0, 1,,, l z l l (By convention, we pick the z-direction be the relevant direction for this quantization. Physically, there are no preference in the z-direction. The other two directions are not quantized.)

64 Magnetic Quantum Number Illustrations showing the relation between L and L z.

65 Zeeman Effect Experimentally, it was found that under a magnetic field, the energy levels of the H-atom will split according to the magnetic quantum number m l. Semi-classical explanation: L B e - e - orbits around the nucleus and it forms a current loop. L z measures the orientation of L with respect to B and thus affects the energy level of the H-atom.

66 Anomalous Zeeman Effect Predicted with alone m l Additional experiments shows that some of the Zeeman lines are further split.

67 Electron Spins In 195, using again semi-classical model, Samuel Goudsmidt and George Uhlenbech demonstrate that this fine structure splitting is due to the spin angular momentum of the electron and this introduces the 4 th quantum number. 4. Spin Quantum Number: The electron has another intrinsic physical characteristic akin to spin angular momentum associated with a spinning top. This quantum characteristic did not come out from Schrodinger s original theory. Its existence requires the consideration of relativistic quantum effects (Dirac s Theory). The direction of the spin angular momentum S z of the electron is quantized: S m, m z s s 1 S s( s1), s m s 1 Pauli and Bohr

68 Wavefunction Labeling Scheme We have identified 4 separate quantum numbers for the H-atom (n, l, m l, m s ). For a given principal quantum number n, the H-atom has a given energy and there might be more than one distinct states (with additional choices for the other three quantum numbers). The fact that there are more than one distinct states for the same energy is call degeneracy. Historically, states with different principal quantum numbers are labeled as: n1: n : n 3: n 4: K shell L shell M shell N shell And, states with different orbital quantum numbers are labeled as: l 0: l 1: l : l 3: l 4: l 5: s subshell p subshell d subshell f subshell g subshell h subshell

69 Wavefunction Labeling Scheme m l and m s are not labeled by this scheme.

70 Many Electron Atoms The Schrodinger equation for the general case with many electrons and protons interacting together quickly becomes very complicated. Central Field Approximation: Consider the effects from all electrons together as a spherically symmetric charge cloud so that each individual electron sees a total E field due to the nucleus + averaged-out cloud of all other electrons, In this approximation, U(r) is spherically symmetric (depends on r instead of all three spatial coordinates) This approximation is useful to understand the ground state of many electron atoms One can continue to use the 4 quantum numbers for the H-atom (n, l, m l, m s ) to describe them.

71 Pauli Exclusion Principle In order to understand the full electronic structures of the all elements beyond the simple single-electron H-atom, we need another quantum idea. In 195, Wolfgang Pauli proposed the Pauli s Exclusion Principle: no two electrons can occupy the same quantum-mechanical state in a given system, i.e., no two electrons in an atom can have the same set of value for all four quantum numbers (n, l, m l, m s ). The Pauli s Exclusion Principle + the set of the four quantum numbers give the complete prescription in identifying the ground state configuration of e - s for all elements in the Periodic Table. Then, all chemical properties for all atoms follow! Additional electrons cannot all crowded into the n = 1 state due to the Pauli s Exclusion Principle and they must distribute to other higher levels according to the ordering of the four quantum numbers.

72 Filling in the Ground State: Example H-atom (Z = 1 one e - ) n = 1, l = 0 E Helium (Z = two e - ) n = 1, l = 0 E 1 filled, 1 free space the lowest level is now full Lithium (Z = 3 three e - ) n =, l = 1 n =, l = 0 n = 1, l = 0 E m l Last electron must go to n=, l=0 level by Pauli s Exclusion Principle. n = level

73 Filling in the Ground State: Example Sodium (Z = 11) m l n = 3, l = 0 n =, l = 1 n =, l = 0 n = 1, l = 0 E

74 Spectroscopic Notation in the Periodic Table Typically, only the outer most shell (including the subshells within the outer most shell) is labeled. # of e - in that subshell H 1s shell n value 1 subshell l value He 1s 4 O 1s s p Z = 8 outer shell is n = two subshells (l =0 and l = 1) s p 8 electrons to fill, will fill K shell and 6 remaining will need to go to L shell: sl ( 0) : ml 0 pl ( 1): ml 1,0,1 only max slots 6 max slots with 4 taken

75 Ground-State Electron Configurations