Quantum Phenomena PDF

Preview

Summary

quantum mechanics of the atom, including energy levels, spin, and other quantum phenomena ...

Description

PHYSICS 1

QUANTUM PHENOMENA II DR ROBERT P CAMERON [email protected] KELVIN 522

INTRODUCTION

THE STRUCTURE OF ATOMS

By 1913, the following facts about the structure of atoms had been established by Thomson, Millikan, Rutherford, Geiger, Marsden and others: An atom has an overall size of ∼ 10m and consists of one or more electrons bound electromagnetically to a single nucleus. The electrons each have mass   9.11  10kg and negative electric charge  1.60  10C. The nucleus has a much smaller diameter of ∼ 10  10 m, a mass  ≫   close to that of the atom as a whole and a positive electric charge that cancels the electric charge of the electron(s) such that the atom is electrically neutral overall. Profound mysteries remained, however. We will focus our attention initially here upon the hydrogen atom, which has one electron. We neglect the finite size and internal structure of the nucleus and approximate  → ∞.

Hydrogen is the most abundant element in the universe. Photograph courtesy of the Hubble space telescope. A classical picture of the hydrogen atom in which the electron orbits the nucleus due to the Coulomb force much as the earth orbits the sun due to the gravitational force fails on at least two counts: ■

The atom is predicted to be unstable as the electron is accelerating and so should radiate electromagnetically, spiralling into the nucleus accordingly on a very short time scale.

■

The picture offers no explanation for the discrete line spectrum of hydrogen.

Enter the Bohr model of the hydrogen atom…

BOHR MODEL OF THE HYDROGEN ATOM In 1913 Bohr presented a radical new model of the hydrogen atom, based upon the following postulated deviations from classical behaviour: ■

The electron can only occupy circular orbits for which    



 !"

where ! ∈ $1,2, … ( labels the allowed orbits,  is the magnitude of the electron’s orbital angular momentum about the nucleus in the !th orbit,  is the radius of the !th orbit and  is the orbital speed of the electron in the !th orbit. Each of these orbits is inherently stable.

■

The electron can make a transition between two of these orbits by absorbing or emitting a photon with energy equal to that gained or lost by the electron.

Bohr postulated the existence of stable orbits for the electron in the hydrogen atom. Consider the !th orbit. The Coulomb force has constant magnitude )  and maintains the orbit as ) 

*   *  4,-  *    ! *.

Using Bohr’s first postulate it follows that







/0 !

where .   4,-  "* /*   5.29  10  m is the Bohr radius and /  * /4,-  0" 3 1/137 is the fine structure constant. The energy 6 of the orbit is therefore 6 

 2

* 



7089 *  4,-   !*

where 89     /32, * -* 0"  1.10  10 : m  is the Rydberg constant for  → ∞. The numerical value 7089  13.6eV where 1eV  1.60  10 J is often written. According to Bohr’s second postulate the electron can make a transition between an allowed orbit with lower energy 6> and an allowed orbit with higher energy 6? by absorbing (!@ → !A ) or emitting (!@ ← ! A)

a photon of wavelength C satisfying

1 1 6?  6 > 1   89 D *  * E C 70 ! @ !A . This is in fact the Rydberg formula, which describes well the discrete line spectrum of hydrogen.

The Bohr model correctly predicted the essential features of hydrogen’s line spectrum. As we have seen, the Bohr model correctly predicted the size (2.  ∼ 10m) of the hydrogen atom and the essential features of hydrogen’s line spectrum, as well as other properties such as the ionisation energy of the hydrogen atom. These were huge advancements. Ultimately, however, the Bohr model was found to fail in certain respects and, in spite of efforts by Sommerfeld and others to extend it, was superseded in 1926 by a more fundamental and accurate description based upon the Schrödinger wave equation.

SCHRÖDINGER DESCRIPTION OF THE HYDROGEN ATOM In the Schrödinger description of the hydrogen atom the electron does not move in circular orbits like in the Bohr model. Rather the electron is described in general by a quantum mechanical wavefunction ΨGH, I, J, KL; a complex function of position H, I, J and time K that satisfies the Schrödinger wave equation M"

NΨGH, I, J, KL P ΨGH, I, J, KL O NK

P is the Hamiltonian operator describing the electron moving in the Coulomb potential energy of the where O nucleus. According to Born |ΨGH, I, J, KL| * gives the probability per unit volume of finding the electron at H, I and J at time K, assuming ΨGH, I, J, KL to be normalised. The position of the electron is otherwise indeterminate if it is not measured. Owing to the spherical symmetry here it is most natural to use spherical coordinates , R and S.

Spherical coordinates are useful for problems with spherical symmetry. It is found in particular that the electron can occupy stationary quantum mechanical states with wavefunctions of the form TℓV ℓ G, R, SL  8 ℓGLWℓV ℓ GR, SL

where 8ℓ GL is related to an associated Laguerre polynomial function and WℓV ℓ GR, SL is a spherical harmonic function (we will explore these in the lectures). The associated energies are 6  

7089 !*

These stationary quantum mechanical states are labelled by the principal quantum number ! ∈ $1,2, … (, the orbital angular momentum quantum number ℓ ∈ $0, … , !  1( and the orbital angular momentum projection quantum number ℓ ∈ $0, … , Xℓ(. We have already met ! in as much as the 6 here are identical to those predicted by the Bohr model. In contrast, ℓ and ℓ are new and are discussed below.

Alternative notations for ! and ℓ are often encountered. Those quantum mechanical states with a given value of ! may be referred to collectively as a shell. !  1 is then the K shell, !  2 is the L shell and so on alphabetically thereafter. This nomenclature is particularly prevalent in x-ray spectroscopy. Those quantum mechanical states with given values of ! and ℓ may be referred to collectively as a subshell. Different letters are assigned to different values of ℓ, with ‘s’ (‘sharp’) corresponding to ℓ  0, ‘p’ (‘principal’) corresponding to ℓ  1, ‘d’ (‘diffuse’) corresponding to ℓ  2, ‘f’ (‘fundamental’) corresponding to ℓ  3 and so on alphabetically thereafter. The values of ! and ℓ may then be listed together as 2p for the !  2 and ℓ  1 subshell, for example. This spectroscopic notation proves particularly useful in describing the electron configurations of more complicated atoms, as we will see. In more detail: TℓV ℓ G, R, SL and 6  satisfy the time independent Schrödinger equation P T ℓV G, R, SL  6T ℓV G, R, SL O ℓ ℓ

where

P O



N* N* " * N* D * ] * ] *E  2  NH NI NJ 4,-

*

*  bH

] I* ] J *

N* 2 N 1 N * 0deR N * N* ] ] Ef  ]  ] D 2 N *  N  * NR* eM!R NR NS * 4,-   *



YZ[ *\ ] Z[ *^ ] Z[_* ` ] a GH, I, J L 2 

"

c

The physically motivated boundary conditions that TℓV ℓG, R, SL be finite everywhere whilst lim T ℓV ℓ G, R, SL  0

i→9

and for example, are satisfied.

TℓV ℓ G, R, SL  TℓV ℓ G, R, S ] 2,L,

QUANTISATION OF ANGULAR MOMENTUM

We have seen that the energy 6 of the electron in the hydrogen atom is quantised in that it can only take on certain values which depend upon the principal quantum number ! as 6  7089 /!* .

Similarly, the orbital angular momentum j of the electron about the nucleus is quantised in that its magnitude  can only take on certain values which depend upon the orbital angular momentum quantum number ℓ as   kbℓGℓ ] 1Ll "

whilst its J component _ can only take on certain values which depend upon the orbital angular momentum projection quantum number ℓ as  _   ℓ" Note that |_ | m  unless ℓ  ℓ  0. Thus j cannot be exactly parallel to the J axis. It is found moreover that that the H component \ and the I component ^ of j are indeterminate. These features can be understood in terms of the uncertainty principle.

To convey the idea that  and _ are well defined whilst \ and ^ are not, one sometimes depicts j as lying somewhere on the surface of a suitable cone for ℓ n 0 or disc for  ℓ  0.

It is not possible to specify all three components of an angular momentum like j simultaneously.

It can be shown rather generally that an angular momentum o in quantum mechanics has magnitude p  kbqGq ] 1Ll "

whilst its J component can only take on the values

p_  r"

where q ∈ $0,1/2,1, … ( is an angular momentum quantum number and r ∈ $0, … , Xq( is an angular momentum projection quantum number. We note parenthetically that freely propagating light possesses angular momenta that do not satisfy this description but which are nevertheless associated with rotational symmetries and are conserved. The choice of J as the quantisation axis is a convention.

SPIN It had been established by 1928 through the work of Stern, Gerlach, Goudsmit, Uhlenbeck, Pauli, Dirac and others that the electron has a spin angular momentum s, which is an intrinsic property independent of the position of the electron in space. It was originally proposed that s be attributed to a rotation of the electron about an internal axis, much as an intrinsic angular momentum is attributable to the rotation of the earth about its internal axis. This picture turned out to be somewhat inaccurate, however. The electron is not simply a spinning sphere of charge, though it is possible to relate s to a circulation of energy in the Dirac field and such pictures may certainly help guide your imagination.

The electron has a spin angular momentum s but is not simply a spinning sphere of charge.

The spin angular momentum quantum number of the electron is e  1/2 and the spin angular momentum projection quantum number can therefore take on the values t  X1/2. The magnitude of s is thus

whilst the J component of s is

√3" 1 1 u  kbeGe ] 1Ll "  vw x ] 1yz "  2 2 2 u_  t"  X

" 2

. If t  1/2 we speak of spin up and if t  1/2 we speak of spin down. The half integral spin (e  1/2) of the electron gives rise to some unusual properties. In particular, the electron must be rotated through 4, (twice) to return it to its original state.

In summary, the complete list of quantum numbers for the electron in the hydrogen atom and their significance are as follows: ■ ■

■

■

■

The principal quantum number ! ∈ $1,2, …( determines the energy 6  708 9 /!* .

The orbital angular momentum quantum number ℓ ∈ $0, … , !  1( determines the magnitude   kbℓGℓ ] 1Ll " of the orbital angular momentum j.

The orbital angular momentum projection quantum number  ℓ ∈ $ 0, … , Xℓ( determines the J component _  ℓ" of the orbital angular momentumj .

The spin angular momentum quantum number e  1/2 determines the magnitude u  kbeGe ] 1Ll " 

√ 3"/2 of the spin angular momentum s.

The spin angular momentum projection quantum number  t ∈ $X1/2( determines the J component u_  t" of the spin angular momentum s.

The stationary quantum mechanical states of the electron in the hydrogen atom can be listed as ! 1 2 2 3 3 3

ℓ 0 0 1 0 1 2

ℓ 0 0 0 , X1 0 0, X1 0, X1, X2

e 1/2 1/2 1/2 1/2 1/2 1/2

t X1/2 X1/2 X1/2 X1/2 X1/2 X1/2

shell K L L M M M

spectroscopic notation 1s 2s 2p 3s 3p 3d

and so on. Additional corrections to the basic Schrödinger description include those due to the finite mass of the nucleus ( →   /G  ] L with  the mass of the proton now), the magnetic moment of the electron and relativistic effects (fine structure), the magnetic moment of the nucleus (hyperfine structure) and coupling of the atom to the electromagnetic vacuum (Lamb shift). The acknowledgement of the spin of the electron is as far as we will go with the calculation of the structure of the hydrogen atom, however.

SCHRÖDINGER DESCRIPTION OF MORE COMPLICATED ATOMS The Schrödinger equation cannot be solved analytically for an atom containing more than one electron. Each electron is simultaneously attracted to the nucleus and repelled by the other electrons: the situation is just too complicated! Various levels of approximation can be entertained, however. The most drastic of these is to neglect the interactions between the electrons such that each electron is only attracted to the nucleus. The stationary quantum mechanical states for each electron then have wavefunctions and associated energies like those found for the hydrogen atom (T ℓV ℓ G, R, S L and 6 ) but with a nuclear charge of | where | is the number of protons in the nucleus, known as the atomic number. A better approximation is to suppose that each electron moves not in the Coloumb potential energy | * /4,-   of the nucleus but rather in some other spherically symmetric potential energy }GL chosen to account, in an average way, for the nucleus and the other electrons. This is known as the central field approximation. It is still appropriate to label the quantum mechanical states of each electron by quantum

numbers !, ℓ, ℓ, e and  t , although the  dependences of the stationary state wavefunctions differ from those found for the hydrogen atom (8ℓGL) and the associated energies depend upon both ! and ℓ.

PAULI EXCLUSION PRINCIPLE AND THE PERIODIC TABLE

The Pauli exclusion principle was established in 1925 and tells us that no two electrons can occupy the same quantum mechanical state. Together with what have we have learned above this suffices for a basic understanding or the periodic table of the elements. The number of protons in an atom defines an element and dictates its place on the periodic table. Hydrogen is the first element on the periodic table, with |  1 proton. Helium is next, with |  2 protons, Lithium follows with |  3 protons and so on, up to at least ununoctium with |  118 protons. The number of neutrons in an atom then defines an isotope. The removal of electron(s) from an atom yields a cation which has a net positive electric charge. The addition of electron(s) to an atom yields an anion which has a net negative electric charge. The electron configuration of an atom may be given in the form 1s 22s22p 6… for example. The normally sized numbers here indicate values of !, the letters indicate values of • and the superscript numbers indicate the number of electrons in each subshell so defined. According to the Pauli exclusion principle a given subshell with its 2ℓ ] 1 distinct values of ℓ can house up to 2G2ℓ ] 1L electrons, as each electron can be spin up (t  1/2) or spin down (t  1/2). The electron configuration given above therefore describes full K and L shells, with all other shells empty. To deduce the ground state electron configuration of an element we assign the electrons to the allowed quantum mechanical states that lie lowest in energy whilst ensuring that no two electrons occupy the same quantum mechanical state, to satisfy the Pauli exclusions principle. For a given value of !, energy increases with ℓ. Thus | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

name of element hydrogen helium lithium beryllium boron carbon nitrogen oxygen fluorine neon sodium magnesium aluminium silicon phosphorous sulphur chlorine argon potassium calcium

chemical symbol H He Li Be B C N O F Ne Na Mg Al Si P S Cl Ar K Ca

ground state electron configuration 1s1… 1s2… 1s22s1… 1s22s2… 1s 22s22p1 … 1s 22s22p2 … 2 2 3 1s 2s 2p … 2 2 4 1s 2s 2p … 1s22s22p5 … 1s 22s22p6 … 1s22s 22p63s1… 1s 22s 22p63s2… 2 2 6 2 1 1s 2s 2p 3s 3p … 2 2 6 2 2 1s 2s 2p 3s 3p … 1s2 2s2 2p63s23p 3… 1s2 2s2 2p63s23p 4… 1s2 2s2 2p63s23p 5… 1s2 2s2 2p63s23p 6… 1s22s22p 63s23p64s 1… 2 2 6 2 6 2 1s 2s 2p 3s 3p 4s …

and so on. Subtleties arise for some of the heavier elements due to crossings of energy levels. This is evident in potassium and calcium, where electrons occupy the 4s subshell whilst the 3d subshell, which lies lower in energy than the 4s subshell in the hydrogen atom but not for these atoms, remains unoccupied. Atoms with full subshells tend to be chemically inert whilst those with nearly full subshells tend to be reactive. Looking at the electron configurations above we see, for example, that helium, neon and argon have full subshells. These elements are indeed chemically inert, being noble gases. Hydrogen, lithium and sodium have noble gas like electron configurations but with one extra electron. These elements are indeed highly reactive, with the latter two being alkali metals. Fluorine has a noble gas like electron configuration minus one electron. This element is also highly reactive, being a halogen. In short, the groups (columns) in the periodic table correspond to atoms with analogous electron configurations in their outermost subshells, and therefore similar chemical properties, whilst the periods (rows) correspond to the filling of successive shells in accord with the Pauli exclusion principle.

Quantum mechanics sheds light upon the periodic table of the elements....