Matter Waves with Example in Quantum Physics I - Lecture Slides | PHYS 486, Study notes of Quantum Physics

Download Matter Waves with Example in Quantum Physics I - Lecture Slides | PHYS 486 and more Study notes Quantum Physics in PDF only on Docsity! Page 1 Physics 486 Lecture 2 Physics 486, Spring ‘07 Lecture 2 x Matter Waves and Classical Models of the Atom What About Matter? Breakdown in classical descriptions of atoms – About the same time that classical descriptions of light were running into problems, classical pictures of matter - specifically the atom – were also in trouble for several reasons, principally: Evidence for wavelike character of electrons, e.g., electron diffraction Problems with classical descriptions of the atom In this lecture, we will explore the evidence for – and the applications of – the wave nature of matter. Wavelike properties of matter We’ve seen in the previous lectures that in the early 1900’s, it became clear from the photoelectric effect, the Compton effect, etc., that – in addition to its obvious wavelike character (e.g., interference phenomena) – light exhibits particle-like characteristics under certain circumstances. In 1923-24, Louis DeBroglie took this one step further and proposed that this “wave- particle” duality characterizes not only light, but matter as well! I won the Nobel Prize for this ‘discovery’ in 1929 Wavelike properties of matter This conclusion was based upon the following insightful correspondences noted by DeBroglie: Electronic (Particle) Energy Light (Wave) frequency Free (unbound) electrons can have any energy Free waves can have any frequency Electrons confined in atoms only have discrete energies (e.g., atomic spectra) Waves confined to cavities only have discrete frequencies Electrons confined in atoms have a NON- ZERO lowest allowed energy Waves confined to cavities have a NON-ZERO lowest allowed frequency ⇔ ⇔ ⇔ Discrete Atomic Emission Spectra Electrons in an atomic potential have only certain energies (more on this shortly!): λ (nm) Atomic hydrogen DeBroglie: Matter Waves Based upon these correspondences, DeBroglie proposed that, like photons, particles having a momentum p behave like a group of “matter” waves having a wavelength given by: λ = h/p In order to estimate the de Broglie wavelengths of elementary particles such as electrons, it is useful to derive a relationship for the de Broglie wavelength of particles having a kinetic energy K. So, from the de Broglie relationship, λ=h/p, we get: For non-relativistic particles: 2 21 2 2 pK mv m = = ( )1/ 22p mK= ( )1/ 22 h mK λ = (v << c) DeBroglie Waves (cont.) Where Eo = moc2 is the rest mass energy of the particle, and E = Eo + K For relativistic particles, the total energy is E2 = Eo2 + c2p2 (*) From (*) we get the following relationship for the momentum: ( )2 2 2 2o oE K E c p+ = + ( ) 1/ 221 2 op E K Kc = + So, from the de Broglie relationship, λ=h/p, we get: ( )1/ 222 o hc E K K λ = + Page 2 Physics 486 Lecture 2 Example: Matter Waves What size wavelengths are we talking about? Consider a photon with energy 3 eV, and therefore momentum p = 3 eV/c. Its wavelength is: Note that the kinetic energy of the electron is different from the energy of the photon with the same momentum (and wavelength): λe = h/pe What is the wavelength of an electron with the same momentum? Compared to the energy of the photon (given above): Same relation for particles and photons! ( ) eV.eV/J. J. )m)(kg.( sJ. m h m pKE 619 24 2931 234 2 22 1088106021 10411 10414101192 106256 22 −− − −− − ×=×÷ ×= ×× ⋅× === λ eVpcE 3== ( ) ( ) nms/ms.c eV seV. p h 4141031041 3 10144 81515 =×××=×⋅×== − − λ Crystals consist of regularly spaced atoms -- a regular array of scattering centers. Electromagnetic waves with wavelengths of a few nanometers (X-rays) can diffract off crystals, producing an interference pattern. electron gun Ni Crystal detector θ In 1927-8, it was shown by Davisson-Germer that ELECTRONS can also diffract from crystals ! Evidence for Matter Waves: Electron Diffraction Let’s analyze this process in more detail: Surprise! Electrons can act like waves ! I( θ) θ Interference peak ! 0 60o Evidence for “Matter Waves”? electron gun Ni Crystal detector θ Matter as Waves: Electron Diffraction Evidence for “Matter Waves”? θφ incident beam scattered beam atomic planes d Path difference (PD) between rays 1 and 2: PD = 2s = 2dcos(90º-φ) = 2dsinφ φ d s s 1 2 1 2 90º-φ Constructive interference between rays 1 and 2 occurs when: PD = 2dsinφ = nλ (n=1,2,3,…) electron gun Ni Crystal detector θ Matter as Waves: Electron Diffraction Let’s take case of Ni in Davisson- Germer experiment: Note from diagram that: φ = (180º - θ)/2 φ d s s 1 2 1 2 90º-φ So, λ = 2dsin65º = 2(0.91)sin65º = 1.65 A I( θ) θ 50º Vo = 54 volts θ φ = 65º n=1 (1st order) d=0.91 Aº º electron gun Ni Crystal detector θ Example: Davisson-Germer Experiment Summarizing the experimental result: I( θ) θ 50º Vo = 54 volts d=0.91 Aº λ = 2dsin65º = 2(0.91)sin65º = 1.65 Aº Is this consistent with the deBroglie wavelength prediction? ( )1/ 22 h mK λ = Vo = 54 volts K = 54 electon-volts (eV) ( ) 34 1/ 231 19 6.63 10 2 9.1 10 54 1.6 10 / J s kg eV J eV λ − − − × − = × × × × × = 1.65 A º Electron Diffraction More evidence for the wave-like nature of electrons was provided by G. P. Thomson, when he demonstrated the diffraction of electrons through a polycrystalline thin film: Example of electron diffraction Interference patterns have now been observed of larger and larger particles, including neutrons, protons, helium atoms, hydrogen molecules, atomic gases, and most recently, C60 molecules! Interfering xenon beams Page 5 Physics 486 Lecture 2 Planetary Model However, this model had trouble explaining the observation of discrete emission and absorption spectra: Classically, it was known that an accelerating charged particle gives off radiation continuously, and since electrons in orbit around a nucleus experience centripetal acceleration, the electronic orbit should collapse quickly and the Rutherford atom should be unstable! Also, it had long been known (throughout the 1800’s particularly) that gases of atoms, when excited electrically, emit radiation only at discrete wavelengths Different atoms gave off different emission spectra λ (nm) Atomic hydrogen (Balmer series) This development motivated the Planetary Model of Ernest Rutherford: Negatively charged electrons orbit positively charged nucleus due to Coulomb force. The frequency of light emitted equals electron’s orbital frequency, flight=fo. +Ze -e F helium Atomic Spectra If an electric discharge is passed through a gas, some of the atoms are excited to higher energy states. Upon returning to their lowest, “ground” energy state (through collisions, for example), the atoms give off their excess energy in radiation. One can experimentally observe the characteristic spectral signature of these atoms by collecting the radiation, then sending it through a spectrometer – comprised chiefly of a prism or grating that separates the emission into its various spectral components – then onto a photographic plate or some other detector: θ Slit Eyepiece Light source Undeflected beam Deflected beam Grating Light shieldCross hairs Collimating lensFocusing lens Balmer Series Wavelength, λ As an example, the emission spectra of atomic hydrogen in the visible light region is shown above. The spectral response of hydrogen was found by J. J. Balmer (in ~ 1885) to be mathematically described by the rather simple formula (note that this ‘discovery’ is numerological, NOT based on a microscopic description): 2 o 23646 4n n n λ = Α − n = 3,4,5,… which can also be written: 12 1 1 1109, 700 4n cm nλ − = −    n = 3,4,5,… Bohr’s Postulates The discrete emission spectra of of elemental gases such as hydrogen were clear evidence against the Thompson model of the atom, which held that neutral atoms consisted of an electronic (negative) charge was embedded in a positively charged material. Based upon the discrete emission spectra Neils Bohr postulated in 1913 that ALL the positive charge, and MOST of the mass, of the atom was concentrated at the center of the atom, comprising a very small fraction of the atomic volume. In this Bohr model, the negative electronic charge was postulated to be in orbit about the positively charged nucleus. To account for the discrete emission spectra, Bohr made 2 additional postulates: (1) The angular momentum of the electron in the atom does NOT have arbitrary (continuous) values, but rather has only discrete values given by: (2) Bohr also postulated that the atom radiated ONLY when jumping from one orbital state to another, Erad = Ei – Ef. We know also from Lecture 1 that this emitted radiation is quantized as PHOTONS, Eph=hf! +Ze -e F / 2L nh nπ= = (n=1,2,3,…) Bohr’s Model Let’s see what comes out of Bohr’s model with these postulates. First, according to Newton’s law, F=ma: (F(r)=dV(r)/dr) Now, the total energy of the electron is given by: +Ze -e F 2 2 2 1 4 o mv eF ma r rπε = = = 21( ) 4 o eV r rπε − = 2 2 1 4 o emv rπε = (SI units) 2 21 1 2 4 o eE K V mv rπε = + = − (*) Using (*), we get for the total energy: 2 2 21 1 1 1 1 2 4 4 2 4o o o e e eE r r rπε πε πε = − = − (**) Note: This is all classical! Bohr’s Model Up to now, the calculation has been entirely classical. But now, Bohr imposed his postulate (1), i.e., the angular momentum is quantized: (n=1,2,3,…) So, Bohr’s postulate (1) implies that the electronic orbits in the atom are quantized in units of the quantity ao, where ao is the “Bohr radius”, i.e., the smallest (fundamental) orbit, and n=1,2,3,… L mvr n= = (n=1,2,3,…) Solving this for v, and plugging into (*) on the last slide, gives the following result for r: 2 2 1 4 o ev m rπε = ⇒ ( ) 2 2 1/ 4 o en mr m rπε = 2 2 2 2 4 o or n a nme πε = = Page 6 Physics 486 Lecture 2 Bohr’s Model Furthermore, plugging this condition for the quantized orbits, (n=1,2,3,…) 2 2 1 4 2n o o eE a nπε = − further suggests that the energies of the electrons bound to the atom are also quantized: Finally, Bohr Postulate (2) suggests that: 2 2 2 1 1 1 4 2rad i f o o f i eE E E a n nπε   = − = −     into the classical energy equation (**), 2 2 2 2 4 o or n a nme πε = = 21 1 2 4 o eE rπε = − (n=1,2,3,…) Bohr’s Model Comparing this to the Balmer formula: This gives a value for the Bohr radius: +Ze -e F and that rad hcE λ =Is consistent with the Einstein formula 2 4 1 3 2 1 109,700 4 2 8H o o o e meR cm hca h cπε ε −= = = 1 2 1 1 1109, 700 4n cm nλ − = −    22 2 1 0.529 A 4 2 o o o o H hea hcR me ε πε π = = = Bohr’s Postulates: Problems However, in spite of the fact that the Bohr model phenomenologically describes certain aspects of the atom, such as the discrete emission spectrum, it had a number of inadequacies that begged a more sophisticated approach: (1). The meaning of the “stationary states” (i.e., those that did not radiate, and their relationship to physical electronic orbits about the nucleus, was uncertain, not well motivated, and not explainable by classical electromagnetism. (2). The meaning of the quantum jumps in energy also has no explanation in the Bohr model. (3). The Bohr model is not successful in describing multi-electron atoms. (4). Although the Bohr model predicts that the ground state of the atom should have finite angular momentum, since the model describes a ground state in which the electron is orbiting the nucleus, it is experimentally known that the ground state of hydrogen has NO angular momentum! +Ze -e F Connection with Quantized Radiation 1 2 1 1 1109, 700 4n cm nλ − = −    n = 3,4,5,… The simplicity of the Balmer formula when written in terms of the inverse wavelength is no accident: It reflects the fact that the radiant energy emitted during the course of an atomic transition between initial (Ei) and final (Ef) states is – as we know - quantized in units of energy called photons, Erad = Ei – Ef = Ephoton and as postulated in 1908 by Albert Einstein, the energy of the photon is inversely proportional to the inverse wavelength: photon hcE hf λ = = Where c is the speed of light, c = 3 x 108 m/s, h is Planck’s constant, h= 4.136 x 10-15 eV-s, and hc = 1240 eV-nm. Combining (*) and (**) gives: 2 1 113.6 4n hcE eV nλ  = = −    13.6 eV is the ground state energy of the hydrogen atom! Correspondence Principle But how do we recover from the Bohr picture the well-known relationship that relates the energy of radiation to the electronic frequency, but NOT Planck’s constant, in macroscopic systems? +Ze -e F (to first order in ∆n/ni, for very large n) Consider the large n limit of Bohr’s result, which is associated with large (macroscopic) orbits of the electron, so that ∆n = ni-nf << ni: From the previous slide, the corresponding radiation frequency is, in this limit, just: 2 2 2 2 1 1 1 1 1H H f i i i nR R n n n n −    ∆ − = − − −           2 1 1 1 2H i i nR n n   ∆ ≅ − − +      32 H i nR n ∆ ≅ 2 2 1 1 H f i hchf hcR n nλ   = = −     ⇒32 H i nR hc n ∆ ≅ 3 2 H i cRf n n ≅ ∆ Correspondence Principle But now compare this result to the orbital frequency: This gives the following result for the orbital frequency: 22 2o Lf mr ω π π = = of f n≅ ∆ where L is the angular momentum. Imposing the quantization condition on the angular momentum, L, and the orbital radius r: ( )22 22 2o o L nf mr m a nπ π = = 3 2 H o cRf n = (Note: Using RH = e2/(4πεo2hcao) and ao = 4πεo2 2/me2) Consequently, in the Bohr model, for large n, the radiated frequency is simply an integer multiple of the orbital electron frequency, as one expects in the classical Rutherford model: For large n, the quantum and classical descriptions must merge!