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2.2 Schrödinger’s wave equation Slides: Video 2.2.1 Schrödinger wave equation introduction Text reference: Quantum Mechanics for Scientists and Engineers Section Chapter 2 introduction Schrödinger’s wave equation Quantum mechanics for scientists and engineers David Miller 2.2 Schrödinger’s wave equation Slides: Video 2.2.2 From de Broglie to Schrödinger Text reference: Quantum Mechanics for Scientists and Engineers Sections 2.1 – 2.2 Schrödinger’s wave equation From de Broglie to Schrödinger Quantum mechanics for scientists and engineers David Miller Electrons as waves de Broglie’s hypothesis is that the electron wavelength is given by h p where p is the electron momentum and h is Planck’s constant h 6.62606957 1034 J s Now we want to use this to help construct a wave equation A Helmholtz wave equation If we are considering only waves of one wavelength for the moment i.e., monochromatic waves we can choose a Helmholtz wave equation d 2 2 2 k with k 2 dz which we know works for simple waves with solutions like sin(kz), cos(kz), and exp(ikz) (and sin(–kz), cos(–kz), and exp(–ikz)) A Helmholtz wave equation In three dimensions, we can write this as 2 2 2 2 2 2 2 k 2 x y z which has solutions like sin(k r), cos(k r), and exp(ik r) (and sin(-k r), cos(-k r), and exp(-ik r)) where k and r are vectors From Helmholtz to Schrödinger With de Broglie’s hypothesis h / p and the definition k 2 / then k 2 p / h p / where we have defined h / 2 so k 2 p 2 / 2 Hence we can rewrite our Helmholtz equation p2 2 2 or 2 2 p 2 From Helmholtz to Schrödinger If we are thinking of an electron, we can divide both sides by its mass mo to obtain 2 2 p2 2mo 2mo But we know from classical mechanics that p2 kinetic energy of electron 2mo and in general Total energy (E )=Kinetic energy + Potential energy (V r ) From Helmholtz to Schrödinger So Kinetic energy = p 2 / 2mo = Total energy (E ) - Potential energy (V r ) 2 2 p2 Hence our Helmholtz equation 2mo 2mo 2 2 E V r becomes the Schrödinger equation 2mo or equivalently 2 2 V r E 2mo Schrödinger’s time-independent equation We can postulate a Schrödinger equation for any particle of mass m 2 2 V r E 2m Formally, this is the time-independent Schrödinger equation Probability densities Born’s postulate is that the probability P r of finding an electron near any specific point r in space 2 is proportional to the modulus squared r of the wave amplitude r 2 r can therefore be viewed as a “probability density” with r called a “probability amplitude” or a “quantum mechanical amplitude” 2.2 Schrödinger’s wave equation Slides: Video 2.2.4 Diffraction by two slits Text reference: Quantum Mechanics for Scientists and Engineers Section 2.3 (first part) Schrödinger’s wave equation Diffraction by two slits Quantum mechanics for scientists and engineers David Miller Young’s slits An opaque mask has two slits cut in it, a distance s apart s Young’s slits We shine a plane wave on the mask from the left s Young’s slits What will be the pattern on a screen at a large distance zo? ? s zo Young’s slits The slits as point sources give an interference pattern Young’s slits The distance from the upper source to point x 2 on the screen is 2 x s / 2 z o xs/2 zo s 2 x s / 2 2 z zo 1 x s / 2 / zo2 2 2 o zo x s / 2 / 2 zo 2 zo x 2 / 2 zo s 2 / 8 zo sx / 2 zo x Young’s slits The distance from the lower source to point x on the screen is x s / 2 2 z 2 o zo x s / 2 / 2 zo x 2 s 2 zo x 2 / 2 zo s 2 / 8 zo sx / 2 zo zo xs/2 Young’s slits For large zo the waves are approximately uniformly “bright” i.e., using exponential waves for convenience 2 2 2 s x exp ik x s / 2 zo exp ik x s / 2 zo2 Using our approximate formulas for the distances gives s x exp i exp ik sx / 2 zo exp ik sx / 2 zo where k zo x 2 / 2 zo s 2 / 8 zo Young’s slits Now exp i exp i 2cos sx so s x exp i exp ik 2 zo sx exp ik 2 z o sx sx exp i cos k exp i cos 2 z z o o so the “intensity” of the beam 2 1 2 s x cos sx / zo 1 cos 2 sx / zo 2 Young’s slits The interference fringes are spaced by d s zo / s ds s zo Young’s slits This allows us to measure small wavelengths d s s / zo ds s zo 2.2 Schrödinger’s wave equation Slides: Video 2.2.6 Interpreting diffraction by two slits Text reference: Quantum Mechanics for Scientists and Engineers Section 2.3 (second part) Schrödinger’s wave equation Interpreting diffraction by two slits Quantum mechanics for scientists and engineers David Miller Young’s slits If the upper slit is blocked – no interference pattern Young’s slits If the lower slit is blocked – no interference pattern