Chapter 3. Wave Properties of Particles

Transcription

1 Chapter 3 Wave Properties of Particles 1

2 The Wave Debate Wave-ists Particle-ists Maxwell s Equations Interference Diffraction E = hf, p = h/ Photoelectric Compton Effect 2

3 3

4 3.1 De Broglie waves A moving body behaves in certain ways as though it has a wave nature 4

5 Particle-like Behavior of Light Planck s explanation of blackbody radiation Einstein s explanation of photoelectric effect 5

6 A photon of light of frequency has the momentum p hv c h Why? 6

7 Recall your last lecture E 2 = (pc) 2 + (mc 2 ) 2 But isn t the (rest) mass of a photon 0? So for a photon, E = (pc) E = hf (photon momentum) pc = hf p = h/ 7

8 Recall relativity But isn t the (rest) mass of a photon = 0? E 2 = (pc) 2 + (mc 2 ) 2 So for a photon E 2 = (pc) 2 E= pc (photon momentum) pc = hf p = h/ But doesn t E = hf also.. 8

9 The wavelength of a photon is therefore specified by its momentum according to the relation Photon wavelength (3.1) h mv 9

10 de Broglie: Suggested the converse All matter, usually thought of as particles, should exhibit wave-like behavior Implies that electrons, neutrons, etc., are waves! Prince Louis de Broglie ( ) 10

11 de Broglie Wavelength Relates a particle-like property (p) to a wave-like property () 11

12 de Broglie 12

13 photon model wave model E = hf p = h 13

14 De Broglie suggested that Eq. (3.1) is a completely general one that applies to material particles as well as to photons. The momentum of a particle of mass m and velocity is p = mv, and its de Broglie wavelength is accordingly De Broglie wavelength h (3.2) mv The greater the particle s momentum, the shorter its wavelength. In Eq. (3.2) is the relativistic factor 1 (3.3) 1 v 2 / c 2 14

15 Wave-Particle Duality particle wave function 15

16 A travelling wave amplitude c 16

17 Models of a free electron electron Wavelength = h/p amplitude electron Momentum p = mv 17

18 Example: de Broglie wavelength of an electron Mass = 9.11 x kg Speed = 10 6 m / sec Joules sec ( kg)(10 m/sec) 6 10 m This wavelength is in the region of X-rays 18

19 Example: de Broglie wavelength of a ball Mass = 1 kg Speed = 1 m / sec Joules sec (1 kg)(1 m/sec) 6 34 m This is extremely small! Thus, it is very difficult to observe the wave-like behavior of ordinary objects. 19

20 Light Matter Waves Particles So Light exhibits both WAVE and PARTICLE behaviour. and matter PARTICLEs can exhibit WAVE behaviour. 20

21 3.2 Waves of what? Waves of probability 21

22 Water waves : height of the water surface Sound waves : pressure Light waves : electric and magnetic field Matter waves : Wave function? 22

23 The wave function itself, however, has no direct physical significance. Probability density The probability of experimentally finding the body described by the wave function at the point x, y, z at the time t is proportional to the value of there at t. 2 This interpretation was first made by Max Born in

24 Born 2 The probability of finding an electron at a given location is proportional to the square of. 24

25 3.3 Describing a wave A general formula for waves 25

26 Phase Velocity How fast is the wave traveling? The phase velocity is the wavelength / period: v = The wave moves one wavelength,, in one period,. x Since = 1/: v = v In terms of the k-vector, k = 2 and the angular frequency, = 2 this is: v = / k It s also helpful to define a phase delay, T, that a wave experiences after propagating a distance, d: T = d / v 26

27 How fast do de Broglie waves travel? If we call the de Broglie wave velocity v p, we can apply the usua l formula v p to find v p. The wavelength l is simply the de Broglie wavelength =h/mv. 27

28 To find the frequency, we equate the quantum expression E=hf with the relativistic formula for total energy E=mc 2 to obtain hf mc 2 f mc h 2 The de Broglie wave velocity is therefore De Broglie phase Velocity Phase velocity is what we have been calling wave velocity p f mc h 2 h mv 2 c particle velocity v < c v p >c????? (3.3) 28

29 Simple harmonic If we choose t=0 when the displacement y of the string at x=0 is a maximum, its displacement at any future time t at the time place is given by the formula x=0 y Acos 2ft (3.4) Figure 3.1 (a) The appearance of a wave in a stretched string at a certain time. (b) How the displacement of a point on the string varies with time. 29

30 y Acos 2ft (3.4) The displacement y of the string at x at any time t is exactly the same as the value of y at x=0 at the earlier time t-x/v p. In eq. (3.4) t t - x/v p Wave formula x y Acos 2f ( t ) (3.5) v p y Acos 2 ( ft fx ) v p x=0 x=v p t y Acos 2 ( ft x ) (3.6) Figure 3.2 Wave propagation. 30

31 Angular frequency = 2 (3.7) Wave number k 2 p (3.8) The unit of is the radian per second and that k is the radian per meter. An angular frequency gets its name from uniform circular motion, where a particle that moves around a circle times per second sweeps out 2 rad/s. The wave number is equal to the number of radians corresponding to a wave train 1 m long, since there are 2 rad in one complete wave. 31

32 Wave formula y Acos( t kx) (3.9) 32

33 3.4 Phase and Group velocities A group of waves need not have the same velocity as the waves themselves 33

34 If the velocity of the waves are the same, the velocity with which the wave group travels is the common phase velocity. However, if the phase velocity varies with wavelength, the different individual waves do not proceed together. This situation is called dispersion. As a result the wave group has a velocity different from the phase velocities of the waves that make it up. This is the case with de Broglie waves. Figure 3.3 A wave group 34

35 Figure 3.4 Beats are produced by the superposition of two waves with different frequencies. 35

36 This phase velocity: Phase and Group Velocities phase t kx const dt k dx 0 The observable is the group velocity (the velocity of propagation of a wave packet or wave group ). Let s consider the superposition of two harmonic waves with slightly different frequencies (>>, k>>k): y1 Acostkx y2 Acos t k k x 2 " " 1 1 cos cos 2cos cos k y y1 y2 2Acos 2t2k kx cos tkx 2 2 k 2Acostkxcos t x The velocity of propagation 2 2 of the wave packet: fast oscillations d envelope = vg -the group velocity within the wave wave group dk 36 group v p dx dt k

37 37 k v p k g v h c m h mc h v m 2 h mv 2 2 k d k d v g Wave number of de Broglie waves Angular frequency of de Broglie waves Group velocity Phase velocity

38 De Broglie group velocity v g v The de Broglie wave group associated with a moving body travels with the same velocity as the body. De Broglie phase velocity v p k the velocity of the body c v 2 v < c v p > c Vp has no physical significance because the motion of the wave group, not the motion of the individual waves that make up the group, corresponds to the motion of the body, and v g < c as it should be. The fact that v p >c for de Broglie waves therefore does not violates 38 special relativity.

39 Material dispersion Waveguide dispersion Material dispersion comes from a frequency-dependent response of a material to waves. For example, material dispersion leads to undesired chromatic aberration in a lens or the separation of colors in a prism. Waveguide dispersion occurs when the speed of a wave in a waveguide, such as a coaxial cable or optical fiber depends on its frequency. This type of dispersion leads to signal degradation in telecommunications because the varying delay in arrival time between different components of a signal "smears out" the signal in time. 39

40 Refraction & Dispersion refraction Short wavelengths are bent more than long wavelengths dispersion Light is bent and the resultant colors separate (dispersion). Red is least refracted, violet most refracted. 40

41 Fiber Dispersion t Fiber Dispersion t 41

42 Light Dispersion Conceptual waves 42

43 The name "dispersion relation" originally comes from optics. It is possible to make the effective speed of light dependent on wavelength by making light pass through a material which has a non-constant index of refraction, or by using light in a non-uniform medium such as a waveguide. In this case, the waveform will spread over time, such that a narrow pulse will become an extended pulse, i.e. be dispersed. In these materials, is known as the group velocity and correspond to the speed at which the peak propagates, a value different from the phase velocity. 43

44 The phase velocity is the rate at which the phase of the wave propagates in space. This is the velocity at which the phase of any one frequency component of the wave will propagate. You could pick one particular phase of the wave and it would appear to travel at the phase velocity. The phase velocity is given in terms of the wave's angular frequency ω and wave vector k by v p k 44

45 The group velocity of a wave is the velocity with which the variations in the shape of the wave's amplitude (known as the modulation or envelope of the wave) propagate through space. The group velocity is defined by the equation v g k where: vg is the group velocity; ω is the wave's angular frequency; k is the wave number. The function ω(k), which gives ω as a function of k, is known 45 as the dispersion relation.

46 Scheme 46

47 Differences in speed cause spreading or dispersion of wave packets 47

48 The group velocity is the speed of the wavepacket The phase velocity is the speed of the individual waves Phase velocity = Group Velocity The entire waveform the component waves and their envelope moves as one. non-dispersive wave. Phase velocity = - Group Velocity The envelope moves in the opposite direction of the component waves. Phase velocity > Group Velocity The component waves move more quickly than the envelope. Phase velocity < Group Velocity The component waves move more slowly than the envelope. Group Velocity = 0 The envelope is stationary while the component waves move through it. Phase velocity = 0 Now only the envelope moves over stationary component waves. 48

49 Dispersion relation In physics, the dispersion relation is the relation between the energy of a system and its corresponding momentum. For example, for massive particles in free space, the dispersi on relation can easily be calculated from the definition of kin etic energy: E 1 2 mv 2 2 p 2m i.e. the dispersion relation in this case is a quadratic function 49

50 Electron Microscope 50

51 An electron microscope 51

52 Diffraction patterns produced by a beam of x-rays and electrons passing through Al foil : X-rays electrons Application: Electron microscopy 52

53 Experimental Evidence for Electron Matter Waves C.J. Davisson and L.H. Germer; G.P. Thomson (1927) Nobel Prize for Physics

54 3.5 Particle Diffraction An experiment that confirms the existence of de Broglie waves 54

55 Davisson-Germer experiment 55

56 (1927) Figure 3.6 The Davisson-Germer experiment. 56

57 Davisson-Germer apparatus 57

58 Scattering of electrons from a crystalline Ni target leads to electron diffraction. 58

59 Davisson and Germer -- VERY clean nickel crystal. Interference is electron scattering off Ni atoms. Ni e e e e e e e e e det. e e e scatter off atoms move detector around, see what angle electrons coming off 59

60 # e s 0 See peak!! 50 0 scatt. angle so probability of angle where detect electron determined by interference of debroglie waves! e e e e e e e det. e e Observe pattern of scattering electrons off atoms Looks like. Wave! Ni 60

61 PhET Sim: Davisson Germer Careful near field view: D = m doesn t work here. For qualitative use only! 61

62 Typical polar graphs of electron intensity after the accident are shown in Fig The method of plotting is such that the intensity at any angle is proportional to the distance of the curve at that angle from the point scattering. 62

63 The first order diffraction maximum (n=1) is usually most intense. Figure 3.7 Results of the Davisson-Germer experiment, showing how the number of scattered electrons varied with the angle between the incoming beam and the crystal surface. The Bragg planes of atoms in the crystal were not parallel to the crystal surface, so the angles of incidence and scattering relative to one family of these planes were both 65 O (see Fig. 3.8). 63

64 Figure 3.8 The diffraction of the de Broglie waves by the target is responsible for the results of Davisson and Germer. 64

65 Reflection Constructive interference 65

66 66

67 Incident x-ray Bragg plane 2d sin n : n 1,2,3, constructive interference Fig X-ray scattering from a cubic crystal. 67

68 (1) Bragg equation for maxima in the diffraction pattern when n 2d sin? d =0.091 nm, =65 O, n= nm sin nm 2 d sin (2) Use de Broglie formula Ignore relativistic consideration h mv 1 KE m 2 2 v KE 54 ev m 0c 0.51MeV mv 2 2mKE kg 54eV J ev h mv 24 kg nm m s Excellent agreement 68!

69 Now we use de Broglie s formula =h/mv to find the expected wavelength of the electrons. The electron kinetic energy of 54 ev is small compared with its rest ene rgy m 0 c 2 of 0.51 MeV, so we can let = mv 2 h mv qv v 2qV m h 2qV m m nm V 2 h 2mqV 2 (9.110 ( For electron 34 ) 2 ) ( ) 150 V m When V=54 (V) ( nm) 69

70 25 O The strong diffracted beam at =50 O and V=54 V arises from wavelike scattering from the family of atomic planes shown, which have a separation distance d=0.91 A. 70

71 The experiment consisted of firing an electron beam from an electron gun on a nickel crystal at normal incidence (i.e. perpendicular to the surface of the crystal). The electron gun consisted of a heated filament that released thermally excited electrons, which were then accelerated through a potential difference of 54 V, giving them a kinetic energy of 54 ev. An electron detector was placed at an angle θ = 50 to obtain a maximum reading, and measured the number of electrons that were scattered at that particular angle. [1] According to the de Broglie relation, a beam of 54 ev had a wavelength of nm. The experimental outcome was nm, which closely matched the predictions of Bragg's law for n =1, θ = 50, and for the spacing of the crystalline planes of nickel (d = nm) obtained from previous X-ray scattering experiments on crystalline nickel. [1] 71

72 Neutron diffraction by a quartz crystal. The peaks present directions in which constructive interference occurred. 72

73 3.6 Particle in a box Why the energy of a trapped particle is quantized 73

74 Figure 3.9 A particle confined to a box of width L. The particle is assumed to move back and forth along a straight line between the walls of the box. 74

75 De Broglie wavelength of trapped particle n 2L n n=1, 2, 3, Figure 3.10 Wave functions of a particle trapped in a box L wide. 75

76 Standing wave n 2L n n 1,2,3 KE E n P 2m n 2 2 h 8mL 2 2 2m h n 2 2 1,2,3 (1) Energy in quantized by n E 1, E 2, E 3 n energy level quantum number (2) since no counterpart in classical physics E 0 Why? If mv 0 then 76

77 Standing waves First Harmonic Standing Wave Pattern Second Harmonic Standing Wave Pattern Third Harmonic Standing Wave Pattern 77

78 Formation of Standing Waves A standing wave pattern is a vibrational pattern created within a medium when the vibrational frequency of the source causes reflected waves from one end of the medium to interfere with incident waves from the source. 78

79 If an upward displaced pulse is introduced at the left end, it will travel rightward across the snakey until it reaches the fixed end on the right side. Upon reaching the fixed end, the single pulse will reflect and undergo inversion. That is, the upward displaced pulse will become a downward 79 displaced pulse.

80 Note that there is a point on the diagram in the exact middle of the medium that never experiences any displacement from the equilibrium position. 80

81 The animation below depicts two waves moving through a medium in opposite directions. The blue wave is moving to the right and the green wave is moving to the left. As is the case in any situation in which two waves meet while moving along the same medium, interference occurs. The blue wave and the green wave interfere to form a new wave pattern known as the resultant. The resultant in the animation below is shown in black. The resultant is merely the result of the two individual waves - the blue wave and the green wave - added together in accordance with the principle of superposition. 81

82 The result of the interference of the two waves above is a new wave pattern known as a standing wave pattern. Standing waves are produced whenever two waves of identical frequency interfere with one another while traveling opposite directions along the same medium. Standing wave patterns are characterized by certain fixed points along the medium which undergo no displacement. These points of no displacement are called nodes (nodes can be remembered as points of no desplacement). The nodal positions are labeled by an N in the animation above. The nodes are always located at the same location along the medium, giving the entire pattern an appearance of standing still (thus the name "standing waves"). A careful inspection of the above animation will reveal that the nodes are the result of the destructive interference of the two interfering waves. At all times and at all nodal points, the blue wave and the green wave interfere to completely destroy each other, thus producing a node. 82

83 Midway between every consecutive nodal point are points which undergo maximum displacement. These points are called antinodes; the anti-nodal nodal positions are labeled by an AN. Antinodes are points along the medium which oscillate back and forth between a large positive displacement and a large negative displacement. A careful inspection of the above animation will reveal that the antinodes are the result of the constructive interference of the two interfering waves. In conclusion, standing wave patterns are produced as the result of the repeated interference of two waves of identical frequency while moving in opposite directions along the same medium. All standing wave patterns consist of nodes and antinodes. The nodes are points of no displacement caused by the destructive interference of the two waves. The antinodes result from the constructive interference of the two waves and thus undergo maximum displacement from the rest position. 83

84 Standing waves mathematically Take two identical waves traveling in opposite directions y 1 = y m sin (kx -wt) y 2 = y m sin (kx + wt) y T = y 1 + y 2 = 2y m cos wt sin kx This uses the identity sin a + sin b = 2cos½(a-b)sin½ (a+b) Positions for which kx = np will ALWAYS have zero field. If kx = np/2 (n odd), field strength will be maximum for particular time 84

85 Standing waves - interpretation y = 2y m cos wt sin kx Positions which always have zero field (kx = np) are called nodes. Positions which always have maximum (or minimum) field (kx = = np/2 (n odd)) are called antinodes. The location of nodes and antinodes don t travel in time, but the amplitude at the antinodes changes with time. 85

86 Standing waves - if ends are fixed If the amplitude must be zero at the ends of the medium through which it travels, then standing waves will only be created if nodes occur at the endpoints. One example is a string with fixed ends, like a violin string Then the wavelength will be some fraction of 2L, where L is the length of the string/antenna/etc. 86

87 Standing waves - if ends are fixed If the amplitude must be zero at the ends of the medium through which it travels, then standing waves will only be created if nodes occur at the endpoints. One example is a string with fixed ends, like a violin string Then the wavelength will be some fraction of 2L, where L is the length of the string/antenna/etc. L=n/2 L=n/2 87

88 Figure 3.11 Energy levels of an electron confined to a box 0.1 nm wide. 88

89 3.7 Uncertainty Principle 1 We cannot know the future because we cannot know the present 89

90 The Uncertainty Principle Classical physics Measurement uncertainty is due to limitations of the measure ment apparatus There is no limit in principle to how accurate a measurement can be made Quantum Mechanics There is a fundamental limit to the accuracy of a measureme nt determined by the Heisenburg uncertainty principle If a measurement of position is made with precision x and a simultaneous measurement of linear momentum is made with precision p, then the product of the two uncertainties can never be less than h/4 90

91 (a) A narrow de Broglie wave group. The position of the particle can be precisely determined, but the wavelength (and hence the particle s momentum) cannot be established because these are not enough waves to measure accurately. x : small : not well defined (b) A wide wave group. Now the wavelength can be precisely determined but not the position of the particle. : well defined x : large Figure

92 Uncertainty principle It is impossible to know both the exact position and exact momentum of an object at the same time. This principle, which was discovered by Werner Heisenberg in 1927, is one of the most significant of physical laws. 92

93 Wave train formed by Fourier transformation Figure 3.13 An isolated wave group is the result of superposing an infinite number of waves with different wavelengths. The narrower the wave group, the greater the range of wavelengths involved. A narrower de Broglie wave group thus means a well-defined position (x smaller) but a poorly defined wavelength and a large uncertainty p in the momentum of the particle the group represents. A wide wave group means a more precise momentum but, a less precise position. 93

94 The Fourier Transform and its Inverse The Fourier Transform and its Inverse: F( ) f( t) exp( it) dt f 1 () t ( )exp( ) 2 F i t d Fourier Transform Inverse Fourier Transform So we can transform to the frequency domain and back. Interestingly, these transformations are very similar. There are different definitions of these transforms. The 2π can occur in several places, but the idea is generally the same. 94

95 The Scale Theorem in action f(t) F() The shorter the pulse, the broader the spectrum! Short pulse t Mediumlength pulse t This is the essence of the Uncertainty Principle! Long pulse t 95

96 Fourier Transform with respect to space If f(x) is a function of position, Fk ( ) f ( x) exp( ikx) dx x Y {f(x)} = F(k) We refer to k as the spatial frequency. k Everything we ve said about Fourier transforms between the t and domains also applies to the x and k domains. 96

97 Pulse Wave group Wave train Gaussian distribution Wave function Fourier transform Figure 3.14 The wave function and Fourier transform for (a) a pulse, (b) a wave group, a wave train, and (d) a Gaussian distribution. A brief disturbance needs a broader range of frequencies to describe it than a disturbance of greater duration. The Fourier transform a Gaussian function is also a Gaussian function. 97

98 ( x ), g ( k ) Gaussian distribution : uncertainty of f ( x0 ) ( x) x ( k) minimum value of k x k x 1 2 k 1 2 Fig.3.14(d) position of uncertainty wave number of uncertainty 98

99 Gaussian function f x x 0 / x e 2 P x x 1 2 x 2 x 1 f 2 x dx P x0 x f xdx x 0 Fig A Gaussian distribution. The probability of finding a value of x is given by the Gaussian function f(x). The mean value of x is x 0 and the total width of the curve at half its maximum value is 2.35, where is the standard deviation of the distribution. The total probability of finding a value of x within a standard deviation of x 99 0 is equal to the shaded area and is 68.3 percent.

100 k x - analogous to minimum bandwidth/minimum pulsewidth 2 hk p h p k 2 2 p h 2 k h p 4 h p x De Broglie wavelength of a particle k x h k x k h x p x h h h p x Uncertainty principle (3.21)

101 H-bar h J s The uncertainty principle becomes xp

102 Figure 3.16 The wave packet that corresponds to a moving packet is a composite of many individual waves, as in Fig The phase velocities of the individual waves vary with their wave lengths. As a results, as the particle moves, the wave packet spreads out in space. The narrower the original wavepacket-that is, the more precisely we know its position at that time -the more it spreads out because it is made up of a greater span of waves with different phase velocities. 102

103 3.8 Uncertainty Principle II A particle approach gives the same result 103

104 Figure 3.17 An electron cannot be observed without changing its momentum. 104

105 The Uncertainty Principle Measurement disturbes the system 105

106 h momentum change of electron?? p h x photon momentum p A reasonable estimate of the minimum uncertainty in the me asurement might be one photon wavelength, so that (3.24) x x p This result is consistent with Eq. (3.22), xp. 2 h (3.25) 106

107 Details 107

108 The act of observation (Compton Scattering) Observations of particle motion by means of scattered illumination. When the incident wavelength is reduced to accommodate the size of the particle, the momentum transferred by the photon becomes large enough to disturb the observed motion. 108

109 Compton Scattering: Shining light to observe electron 109

110 Act of watching: A Through Experiment 110

111 Diffraction by a circular aperture (Lens) 111

112 Resolving Power of Light thru a Lens Image of 2 separate point sources formed by a converging lens of diameter d, ability to resolve them depends on and d because of the inherent diffraction in image formation. 112

113 Putting it all togethor: act of observing an electron 113

114 3.9 Applying the uncertainty principle A useful tool, not just a negative statement 114

115 It is worth keeping in mind that the lower limit of for xp is rarely attained. xp More usually, or even (as we just saw) xp h / 2 115

116 Example 3.7 A typical atomic nucleus is about 5.0x10-15 m in radius. Use the uncertainty principle to place a lower limit on the energy an electron must have if it is to be part of a nucleus. Solution r p 2x m kg m/s r x r If this is the uncertainty in a nuclear electron s momentum, the momentum p itself must be at least comparable in magnitude. An electron with such a momentum has a kinetic energy KE many times greater than its rest energy m o c 2 (=0.51 MeV). KE pc J 20 MeV KE 20 MeV 20 MeV electron is never found! Electron can not exit in a nuclear!! 116

117 Ex. 3.8 A hydrogen atom is 5.3 x10-11 m in radius. Use the uncertainty principle to estimate the minimum energy an electron can have in this atom. Solution m x p 2 x kg m/sec An electron whose momentum is of this order of magnitude behaves like a classical particle, and its kinetic energy is KE 2 p 2m J which is 3.4 ev. The kinetic energy of an electron in the lowest energy level of a hydrogen atom is actually 13.6 ev. 117

118 Energy and time We might wish to measure the energy E emitted during the time in terval t inan atomic process. If the energy is in the form of em waves, the limited time available restricts the accuracy with which we can determine the frequency of the waves. Let us assume that the minimum uncertainty in the number of waves we count in a wave group is one wave. Since the frequency of the waves under study is equal to the number of them we count divided by the time interval, the uncertainty in our frequency measurement is 1 t In general N t 118

119 1 t E E h h t Uncertainty in energy and time E t 2 (3.26) 119

120 Example 3.9 An excited atom gives up its excess energy by emitting a photon of characteristic frequency, as described in Chap. 4. The average period that elapses between the excitation of an atom and the time it radiates is 1.0x10-8 s. Find the inherent uncertainty in the frequency of the photon. Solution t 10 8 sec h The photon energy is uncertain by the amount E 2t E J The corresponding uncertainty in the frequency of light is Natural linewidth E h Hz 120

121 Energy-time uncertainty relation Et Transitions between energy levels of atoms are not perfectly sharp in frequency. n = 3 An electron in n = 3 will spontaneo E h 32 usly decay to a lower level after a n = 2 lifetime of order t~10-8 s. /2 n = 1 There is a corresponding spread in the emitted frequency Intensity Frequency

122 This is the irreducible limit to the accuracy with which we can determine the frequency of the radiation emitted by an atom. As a result, the radiation from a group of excited atoms does not appear with the precise frequency. For a photon whose frequency is, say 5.0x10 14 Hz, /=1.6x10-8. In practice, other phenomena such as the doppler effect contribute more than this to the broadening of spectral lines. 122

123 Doppler broadening In atomic physics, Doppler broadening is the broadening of spectral lines due to the Doppler effect in which the thermal movement of atoms or molecules shifts the apparent frequency of each emitter. The many different velocities of the emitting gas result in many small shifts, the cumulative effect of which is to broaden the line. The resulting line profile is known as a Doppler profile. The broadening is dependent only on the wavelength of the line, the mass of the emitting particle and the temperature, and can therefore be a very useful method for measuring the temperature of an emitting gas. 123

124 Line Broadening Molecular absorption takes place at distinct wavelengths (frequencies, energy levels) Actual spectra feature absorption bands with broader features 1. Natural line width from uncertainty principle (very small) (Lorentzian line shape) 2. Pressure (collisional) broadening (Lorentzian line shape) 3. Doppler broadening (Gaussian line shape) 124