DeBroglie Hypothesis Problem with Bohr Theory: WHY L = n ? have whole numbers with standing waves: n( /2) = L consider roundabout way to stand wave: n = 2 r from Bohr hypothesis: L = mvr = nh/2 Re-organize to get 2 r = nh/mv = n Therefore, n = 2 r = nh/mv, which prompts to = h/mv = h/p .Slide 2
DeBroglie Hypothesis DeBroglie = h/mv = h/p For this situation, we are thinking about the electron to be a WAVE , and the electron wave will "fit" around the circle if the force (and vitality) is perfect (as in the above connection). Be that as it may, this will happen just for particular cases - and those are the particular permitted circles (r n ) and energies (E n ) that are permitted in the Bohr Theory!Slide 3
DeBroglie Hypothesis The Introduction to Computer Homework on the Hydrogen Atom (Vol. 5, number 5) demonstrates this electron wave fitting around the circle for n=1 and n=2. What we now have is a wave/molecule duality for light (E&M versus photon), AND a wave/molecule duality for electrons !Slide 4
DeBroglie Hypothesis If the electron carries on as a wave, with = h/mv , then we ought to have the capacity to test this wave conduct through obstruction and diffraction. Truth be told, tests demonstrate that electrons DO EXHIBIT INTERFERENCE when they experience various openings, similarly as the DeBroglie Hypothesis shows.Slide 5
DeBroglie Hypothesis Even neutrons have demonstrated impedance marvels when they are diffracted from a precious stone structure as indicated by the DeBroglie Hypothesis: = h/p . Take note of that h is little, so that typically will likewise be little (unless the mv is additionally little). A little implies next to no diffraction impacts [1.22 = D sin( )].Slide 6
Quantum Theory What we are currently managing is the Quantum Theory : molecules are quantized (you can have 2 or 3, yet not 2.5 iotas) light is quantized (you can have 2 or 3 photons, yet not 2.5) also, we have quantum numbers (L = n , where n is a whole number)Slide 7
Heisenberg Uncertainty Principle There is a noteworthy issue with the wave/molecule duality: a wave with a positive recurrence and wavelength (e.g., a pleasant sine wave) does not have a distinct area. [At an unmistakable area at a particular time the wave would have a distinct stage, however the wave would not be said to be found there.] [ a pleasant voyaging sine wave = A sin(kx- t) ]Slide 8
Heisenberg Uncertainty Principle b) A molecule has an unequivocal area at a particular time, yet it doesn\'t have a recurrence or wavelength. c) Inbetween case: a gathering of sine waves can include (through Fourier examination) to give a semi-unmistakable area: an aftereffect of Fourier investigation is this: the more the gathering appears as a spike, the more waves it takes to make the gathering.Slide 9
Heisenberg Uncertainty Principle An unpleasant drawing of a specimen inbetween case, where the wave is to some degree confined, and made up of a few frequencies.Slide 10
Heisenberg Uncertainty Principle A formal proclamation of this (from Fourier investigation ) is: x * k (where k = 2 / , and shows the vulnerability in the esteem). Yet, from the DeBroglie Hypothesis, = h/p , this vulnerability connection gets to be: x * (2 / ) = x * (2 p/h) = 1/2 , or x * p = /2 .Slide 11
Heisenberg Uncertainty Principle x * p = /2 The above is the BEST we can do, since there is constantly some exploratory instability. Accordingly the Heisenberg Uncertainty Principle says: x * p > /2 .Slide 12
Heisenberg Uncertainty Principle A comparable connection from Fourier examination for time and recurrence: t * = 1/2 prompts to another piece of the Uncertainty Principle ( utilizing E = hf ): t * E > /2 . There is a third part: * L > /2 (where L is the precise energy esteem). The majority of this is an immediate aftereffect of the wave/molecule duality of light and matter.Slide 13
Heisenberg Uncertainty Principle Let\'s take a gander at how this functions by and by . Consider attempting to find an electron some place in space. You may attempt to "see" the electron by hitting it with a photon. The following slide will demonstrate a glorified chart, that is, it will demonstrate an outline accepting a distinct position for the electron.Slide 14
Heisenberg Uncertainty Principle We fire an approaching photon at the electron, have the photon hit and bob, then follow the way of the active photon back to see where the electron was. approaching photon electronSlide 15
Heisenberg Uncertainty Principle screen opening so we can decide bearing of the active photon active photon electronSlide 16
Heisenberg Uncertainty Principle Here the wave-molecule duality makes an issue in figuring out where the electron was. photon hits here opening so we can decide heading of the active photon electronSlide 17
Heisenberg Uncertainty Principle If we improve the opening smaller to decide the course of the photon (and subsequently the area of the electron, the wave way of light will bring about the light to be diffracted. This diffraction example will bring about some vulnerability in where the photon really originated from, and henceforth some instability in where the electron was .Slide 18
Heisenberg Uncertainty Principle We can diminish the diffraction point on the off chance that we lessen the wavelength (and thus increment the recurrence and the vitality of the photon). In any case, in the event that we do build the vitality of the photon, the photon will hit the electron harder and make it move more from its area, which will expand the vulnerability in the force of the electron.Slide 19
Heisenberg Uncertainty Principle Thus, we can diminish the x of the electron just to the detriment of expanding the vulnerability in p of the electron.Slide 20
Heisenberg Uncertainty Principle Let\'s consider a moment case : attempting to find an electron\'s y position by making it experience a tight opening: just electrons that endure the limited opening will have the y esteem decided inside the instability of the opening width.Slide 21
Heisenberg Uncertainty Principle But the more we tight the opening ( diminish y ), the more the diffraction impacts (wave viewpoint), and the more we are dubious of the y movement ( increment p y ) of the electron.Slide 22
Heisenberg Uncertainty Principle Let\'s investigate how much vulnerability there is: x * p > /2 . Take note of that /2 is a modest number (5.3 x 10 - 35 J-sec).Slide 23
Heisenberg Uncertainty Principle If we were to apply this to a steel chunk of mass .002 kg +/ - .00002 kg, moving at a speed of 2 m/s +/ - .02 m/s, the vulnerability in energy would be 4 x 10 - 7 kg*m/s . From the H.U.P, then, as well as could be expected make certain of the position of the steel ball would be: x = 5.3 x 10 - 35 J*s/4 x 10 - 7 kg*m/s = 1.3 x 10 - 28 m !Slide 24
Heisenberg Uncertainty Principle As we have quite recently illustrated, the H.U.P. becomes an integral factor just when we are managing little particles (like individual electrons or photons), not when we are managing typical size items!Slide 25
Heisenberg Uncertainty Principle If we apply this standard to the electron circumventing the particle, then we know the electron is some place close to the molecule, ( x = 2r = 1 x 10 - 10 m) then there ought to be in any event some instability in the energy of the iota: p x > 5 x 10 - 35 J*s/1 x 10 - 10 m = 5 x 10 - 25 m/sSlide 26
Heisenberg Uncertainty Principle Solving for p = mv from the Bohr hypothesis [KE + PE = E add up to , (1/2)mv 2 - ke 2/r = - 13.6 eV gives v = 2.2 x 10 6 m/s ] gives p = (9.1 x 10 - 31 kg) * (2.2 x 10 6 m/s) = 2 x 10 - 24 kg*m/s; this implies p x is between - 2 x 10 - 24 kg*m/s and 2 x 10 - 24 kg*m/s, with the base p x being 5 x 10 - 25 kg*m/s, or 25% of p.Slide 27
Heisenberg Uncertainty Principle Thus the H.U.P. says that we can\'t generally know precisely where and how quick the electron is circumventing the particle at a specific time. This is predictable with the possibility that the electron is really a wave as it moves around the electron.Slide 28
Quantum Theory But in the event that an electron goes about as a wave when it is moving, WHAT IS WAVING ? At the point when light goes about as a wave when it is moving, we have recognized the ELECTROMAGNETIC FIELD as waving. Be that as it may, attempt to review: what is the electric field ? Can we straightforwardly gauge it?Slide 29
Quantum Theory Recall that by definition, E = F/q . We can just discover that a field exists by measuring an electric drive! We have turned out to be so used to working with the electric and attractive fields, that we tend to underestimate their reality. They absolutely are a helpful build regardless of the possibility that they don\'t exist.Slide 30
Quantum Theory We have four LAWS overseeing the electric and attractive fields: MAXWELL\'S EQUATIONS . By consolidating these laws we can get a WAVE EQUATION for E&M fields, and from this wave condition we can get the speed of the E&M wave and considerably more, (for example, reflection coefficients, and so forth)Slide 31
Quantum Theory But what do we accomplish for electron waves? What laws or new law would we be able to find that will work to give us the abundance of prescient power that MAXWELL\'S EQUATIONS have given us?Slide 32
Quantum Theory The way you get laws is attempt to clarify something you definitely think about, and afterward check whether you can sum up. An effective law will clarify what you definitely think about, and anticipate things to search for that you may not think about. This is the place the legitimacy (or possibly handiness) of the law can be affirmed.Slide 33
Quantum Theory Schrodinger began with the possibility of Conservation of Energy : KE + PE = E add up to . He noticed that KE = (1/2)mv 2 = p
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