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W.N. Catford/P.H. Regan 1AMQ 59 Last Lecture t V x (- 2 2 2 2m ) i This is the Schrödinger equation. i.e. The probability that particle is somewhere = 1 Since k (the wave number) is real, we have shown that any positive value of energy is allowed for a free particle. FREE PARTICLE When a particle is confined by a potential then the solutions of the Schrodinger eqn. are quantised.l

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W.N. Catford/P.H. Regan 1AMQ 60 1AMQ, Part III The Hydrogen Atom 5 Lectures Spectral Series for the Hydrogen Atom. Bohr’s theory of Hydrogen. The Hydrogen atom in quantum mechanics. Spatial quantization and electron spin. Fine Structure and Zeeman splitting. K.Krane, Modern Physics, Chapters 6 and 7 Eisberg and Resnick, Quantum Physics, Chapters 4, 7 & 8

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W.N. Catford/P.H. Regan 1AMQ 61 Atomic Line Spectra Light emitted by free atoms has fixed or discrete wavelengths. Only certain energies of photons can occur (unlike the continuous spectrum observed from a Black Body). Atoms can absorb energy (become excited) by collisions, fluorescence (absorption and re- emission of light) etc. The emitted light can be analysed with a prism or diffraction grating with a narrow collimating slit (see figure below). The dispersed image shows a series of lines corresponding to different wavelengths, called a line spectrum. (note you can have both emission and absorption line spectra)

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W.N. Catford/P.H. Regan 1AMQ 62 Hydrogen-The Simplest Atom Atomic hydrogen (H) can be studied in gas discharge tubes. The strong lines are found in the emission spectrum at visible wavelengths, called H and . More lines are found in the UV region, more which get closer and closer until a limit is reached.

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W.N. Catford/P.H. Regan 1AMQ 63 These lines are also seen in stellar spectra from absorption in the outer layers of the stellar gas.

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W.N. Catford/P.H. Regan 1AMQ 64 The visible spectrum lines are the Balmer Series. Balmer discovered that the wavelengths of these lines could be calculated using the expression, where, R H = the Rydberg constant for hydrogen = 1.097x10 7 m -1 = 1/911.76 angstroms Balmer proposed more series in H with wavelengths given by the more general expression, n 1 >n 2 for positive integers. These series are observed experimentally and have different names n 2 = 1 Lyman Series (UV) = 2 Balmer Series (VIS) = 3 Paschen Series (IR) = 4 Bracket Series (IR) = 5 Pfund Series (IR)

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W.N. Catford/P.H. Regan 1AMQ 65

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W.N. Catford/P.H. Regan 1AMQ 66 These lines are also seen in stellar spectra from absorption in the outer layers of the stellar gas.

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W.N. Catford/P.H. Regan 1AMQ 67 The Rydberg-Ritz Combination Principle is an empirical relationship which states that if and 2 are any 2 lines in one series, then | is a line in another series. Electron Levels in Atoms. Discrete wavelengths for emis/abs. lines suggests discrete energy levels. Balmer’s formula suggests that the allowed energies are given by (cR H / n 2 ) (for hydrogen). A more detailed study is possible using controlled energy collisions between electrons and atoms such as the Franck-Hertz experiment. Excitation Energy ground state, (ie. lowest one) 1 st excited state e - s with enough energy can cause this transition.

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W.N. Catford/P.H. Regan 1AMQ 68 Franck-Hertz Experiment – (1914) Tube filled with gas being studied. Gas is at low pressure. Grid Collector Heated Cathode ● Electrons are emitted by the heated filament. ● They are accelerated by the potential difference V between the cathode and grid. This gives them an energy ½ mv 2 = eV ● If they lose no energy in collisions with the gas atoms/molecules then they pass through the grid and reach the collector and produce a current in the circuit.

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W.N. Catford/P.H. Regan 1AMQ 69 Franck-Hertz Experiment – (1914) Results for Hg ( Mercury) Plot shows the current as a function of the accelerating voltage. The current gradually increases with the potential V since there Is little interaction with the gas other than elastic scattering. At 4.88 V there is a sudden drop in current due to inelastic collisions between the electrons and atoms in the vapour in which the electrons give up kinetic energy in exciting the atoms from their ground states to the first excited state. The current again increases until the electron has a high enough initial energy to excite two atoms in two successive Collisions. Then it is three atoms excited etc. Now 1 st line of Principal series in Hg has λ = 253.7 nm h = hc/λ = 7.83 x 10 -19 J = 4.88 eV We have confirmation that radiation is quantised.

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W.N. Catford/P.H. Regan 1AMQ 70

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W.N. Catford/P.H. Regan 1AMQ 71 Bohr-Rutherford Atom This first successful model was based on the experiments by Geiger and Marsden ( 1911) carried out at Rutherford’s behest. It assumes a central, massive, positively charged(+Ze) nucleus With negative electrons in orbit around it. Classically this would not work. Any charged particle which is Accelerated emits radiation. This is called bremsstrahlung. Thus the electron in orbit, which is accelerated towards the Centre, would lose energy and could not maintain its orbit[just Like an orbiting Earth satellite] Bohr introduced a series of postulates to take account of this.

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W.N. Catford/P.H. Regan 1AMQ 72 Bohr-Rutherford Model Bohr’s Quantum postulates:- 1.Only certain discrete orbits and energies are allowed for the electron--stationary states. 2.Classical mechanics can be used to describe electrons in these orbits 3.A quantum jump between two allowed states releases energy in a single quantum. 4.The allowed orbits are determined by the quantisation of angular momentum. L = r x p Thus mv 2 r = Ze 2 4 0 r 2 and mvr n = nh/2 = mv x r Where n takes integer values 1,2,3,4,5,--------

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W.N. Catford/P.H. Regan 1AMQ 73 Bohr Theory: Bohr’s postulates defined a simple ordered system for the atom.

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W.N. Catford/P.H. Regan 1AMQ 74 2 0 2 2 4 ).( then: 2 Since: nnen e nnen r Ze rm n r m n h nLrm v a 0 is the Bohr radius (i.e. smallest allowed) For hydrogen, Z=1 and a 0 =0.529Angstroms, i.e. model predicts ~10 -10 m for atomic diameter. For these allowed radii, we can calculate the allowed energies of the levels in the Bohr atom K r Ze E r v m r r v mVKE n n n ne n n nen 0 2 2 2 0 2 0 2 2 4. 2 1 thus, for circle 4 and 42 1

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W.N. Catford/P.H. Regan 1AMQ 75 Also, from the circle equation, we can obtain :.....3,2,1, 1. 2)2)2 4( obtain, we,foronsubstitutiby...3,2,1, 4 1 and...4,3,2,1, 4 thus, 4 Simplifying to:.44 22 0 42 2 0 2 22 0 22 0 2 2 0 2 0 2 n n eZm E r n n Ze rm n v n m n r rm n rm n rmr v m e n n ne n e n ne ne nenne ie. quantization of the angular momentum leads in the Bohr model to a quantization of the allowed energy states of the atom.

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W.N. Catford/P.H. Regan 1AMQ 76 For H, the ground state energy is given by eV n E JE eZm En n e 2 18- 1 22 0 42 1 6.13 and 13.62.18x10 i.e. 1 1. 2)2)2 4( thus,,1

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W.N. Catford/P.H. Regan 1AMQ 77 rel).-(non 1 0.73% H,,1for. 1 10x19.2 16 0 2 ni.e ms anm n v e n The allowed energies let us calculate the allowed frequencies for photons emitted in transitions between different atomic levels i.e. h E initial - E final If n i and n f are the quantum numbers of the initial and final states, then, for H, if n f < n i Å 27911. 1 )4(4 thus, 11. 11. 6.13 32 0 4 2222 c em R nn cR nnh eV e H fi H fi This prediction of the Bohr model compares with the experimental value of 1/911.76Å i.e. accurate to within 0.05% in H. (Exact agreement if motion of nucleus is included, i.e. nucleus and electron move around the atom’s centre of mass.) Balmer series corresponds to when n f = 2 Lyman series corresponds to when n f = 1

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W.N. Catford/P.H. Regan 1AMQ 78 The Hydrogen Atom in Quantum Mechanics The e- is bound to the nucleus (p) by the Coulomb potential. This constraint leads to energy quantization. The Time Independent Schrödinger Equation can be used. 222 2 0 2 2 2 2 2 22. 4 1 ),,( where, ),,( 2 zyx e zyxV EzyxV dz d dy d dx d m For hydrogen, (Z=1) It is easier to solve this in spherical polar rather than cartesian coordinates, and for this we have r e rVErV d d rd d rdr d r d m 2 0 2 2 2222 22 4 1 ),,( where,),,( sin 1 )(sin sin 12 2 This equation is said to be separable ie, z x y r

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W.N. Catford/P.H. Regan 1AMQ 79 The Lyman series corresponds to the highest energy (shortest wavelength) transitions which H can emit.

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W.N. Catford/P.H. Regan 1AMQ 80 Deficiencies of the Bohr Model No proper account of quantum mechanics (de Broglie waves etc.) It is planar and the “real world” is three dimensional. It is for single electron atoms only. It gets the predicted angular momenta wrong by one unit of h/2 (experiment shows that in the lowest state of hydrogen, the electron has zero units of orbital Angular momentum, and not one unit, etc.)

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