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Physics 214

Physics 214. 5: Quantum Physics. Photons and Electromagnetic Waves The Particle Properties of Waves The Heisenberg Uncertainty Principle The Wave Properties of Particles A Particle in a Box The Schroedinger Equation The Simple Harmonic Oscillator. Photons and Electromagnetic Waves.

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Physics 214

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  1. Physics 214 5: Quantum Physics • Photons and Electromagnetic Waves • The Particle Properties of Waves • The Heisenberg Uncertainty Principle • The Wave Properties of Particles • A Particle in a Box • The Schroedinger Equation • The Simple Harmonic Oscillator

  2. Photons and Electromagnetic Waves • Long wavelength electromagnetic radiation acts as “waves” • consider a radio wave with = 2.5 Hz • E = ~10-8 eV too small be detected as a single photon...... • a detectable single would require ~1010 such photons, which on the average act as a continuous wave • Short wavelength electromagnetic radiation acts as “particles” • consider a X ray wave with = 1018 Hz • E = ~103 eV, which can easily be detected as a single photon

  3. How should we think of light?

  4. Remember Beats

  5. A solitary pulse can be produced by mixing together waves of infinitely many different harmonic waves of different frequencies, such pulses are called wave-packets. These pulses exhibit properties of both particles and waves • particle • localized • finite size • wave • delocalized Photon

  6. Particles Behaving Like Waves

  7. The wavepacket as whole moves with a velocity vG ---the group velocity The waves in the wavepacket move with a velocity vp----the phase velocity

  8. BEATS ( ) ( ) ( ) y x , t = y x , t + y x , t total 1 2 ( ) ( ) ( ) y x , t = A sin k x - w t + f + A sin k x - w t + f 1 1 1 1 2 2 2 2 total let A = A = A ; f = f = 0 1 2 1 2 [ [ ] [ ] ] [ ] æ æ k - k ö k + k ö w - w w + w ç ç ÷ ÷ t x y = 2 A cos x + sin - t 2 1 1 2 2 1 1 2 2 2 2 2 total è ø è ø ( ) ( ) = 2 A cos k x + w t sin k x - w t 1 4 4 4 2 4 4 4 3 4 4 2 4 4 3 1 b b s s modulated amplitude interference wave beat w - w D w D k w = = ; k = 2 1 2 2 2 b b Beats

  9. Dispersion

  10. Different Phase and Group Velocities

  11. Intensity of Particle Streams and Probability

  12. 1 ( ) 2 u r , t = e , ( ) The instantaneous energy density of a light wave E r t 0 2 E If the frequency of the light is n = , (Eis energy ) this must h ( ) u r , t correspond to photons per unit volume at that point and time E ( ) Thus the photon density at r , t is proportional to the square ( ) of the amplitude of the electromagnetic wave amplitude at r,t

  13. x the wave If there is only one photon, ( ) packet model implies that the photon density at r , t has to be interpreted as the probability that a photon is at the position r at the time t . This interpretation also works even if there are many photons

  14. A wave-packet has an average position, which corresponds to the average position of the photon ò x = x Prob( x ) dx but has no exact position that can be measured The width of the wave-packet corresponds to the dispersion of the positions of the photons ò ( ) ( ) 2 2 D x = x - x = x - x Prob( x ) dx

  15. Prob Noting that ( ) 2 x , t µ E ( x , t ) Prob and the normalization condition for probability distributions ò ( ) x , t dx = 1 we can define 2 E ( x , t ) ( ) Prob x , t = ò 2 E ( x , t ) dx

  16. The Heisenberg Uncertainty Principle A wave-packet has width x It is made up with a range  of wavelength waves D l = l - l = range of wavelengths of max min harmonic oscillators making up wave-packet º proportional to the dispersion of the EXPECTED observed value of a wavelength in the wave-packet º UNCERTAINTY in the observed value of l D l i . e l » l ± 2

  17. D x = x - x = range of spatial positions covered max min by wave-packet º proportional to the dispersion of the EXPECTED observed value of the position of the wave-packet º UNCERTAINTY in the observed value of x D x x » x ± 2 The difference of the wavelengths, Dl , of the waves contained in the wave-packet cannot be greater than the width of the wave-packet , otherwise waves with wavelengths larger than that of the wave-packet would be in the wave-packet hence Þ D x ³ D l

  18. h D l = D p h Þ D x ³ D p Þ D x D p ³ h Heisenberg's Uncertainty Principle A more accurate analysis shows that h h D x D p ³ = 2 4 p æ m D x p D p D x ö æ æ h h h ö ö ç ( ) ( ) ( ) Þ ³ Û D E ³ Û D t D E ³ è ø è ø è ø p m 2 v 2 2 p p D p D x é ù 2 E = Þ D E = & v = ë 2 m m D t û h i . e . D t D E ³ 2

  19. Complex Number Representation of Waves Complex number definition i = -1 i = 2 -1 z = x + iy ( ) ( ) ( ) z ´ z = x + iy ´ x + iy = x x + i y x + x y - y y 1 2 1 1 2 2 1 2 1 2 1 2 1 2 ( ) = x x - y y + i y x + x y 1 1 4 2 2 4 1 3 2 1 2 1 2 1 4 4 2 4 4 3 x iy 3 3 complex conjugate z = x - iy 2 magnitude squared z + = z = z x 2 y 2

  20. e i q = cos q + i sin q 1 ( ) Þ e - e = sin q i - i q q 2 i 1 ( ) Þ e + = cos q e q - i i q 2 Thus E ( ) ( ) ( ) E k , t = E sin kx - w t = e - e ( ) ( ) max - w - - w kx t kx t i i max 2 i

  21. The wave-packet is a linear combination (integral ) of infinitely many waves, thus has a wave-function 2 p é æ ù ö ( ) ( ) E l , x , t = E l sin x - w ( l ) t ê ú è ø ë max l û in general is a complex valued function and its magnitude squared at the point x at time t ( ) ( ) ( ) ( ) 2 Y Y x , t = Y x , t x , t = Prob x , t when K is chosen so that ò 2 ( ) Y x , t dx = 1

  22. The Wave Properties of Particles For massless particles using The Special Theory of Relativity E E = pc Þ p = c and Plancks Hypothesis hc E h E = h n = Þ = l c l h h hk \ p = Û l = & p = = kh l p 2 p de Broglie hypothesized that this was valid for particles with mass also Þ h h l = = & p = kh p m v g

  23. comparison of group velocity to phase velocity for a free particle with mass E n = h E v = ln = p m v g p 2 for a free particle E = 2 m 1 m v 2 v g 2 g Þ ln = = m v 2 g v g \ v = 2 p

  24. de Broglies explanation of the Bohr model using matter waves Electrons are waves, but they are restricted to the one dimensional Bohr orbits They only way they can exist in such a restricted region of space is as standing wave patterns In order to fit into orbits without destructive interference one has to have an integer number of standing wave patterns in one orbit h \ 2 p r = n l = n v m h v Þ m r = n = nh 2 p

  25. This condition is exactly Bohrs Angular Momentum Quantization!!!! Localization of waves Standing waves Þ Þ Quantized energies Electron diffraction from Nickel crystals confirmed de Broglies ideas

  26. The diffraction pattern produced by electrons passing through 2 slits can be viewed as the probability distribution of the electrons hitting the screen • If  << distance between slits then • Slits act as single slits • If  << slit width then • Electrons act as particles • If  ~ slit width or  > slit width electron acts as WAVE !!!! photons display the same behavior

  27. A Particle in a Box U=0

  28. 1 3 L = l = l = l = 2 l 2 1 2 2 3 4 n 2 L L = l Û l = 2 n n n Stationary States for Electron in Box n p æ ö ( ) ( ) ( ) ( ) Y x , t = A sin k x cos w t = A sin x cos w t è ø n n L 2 p n p as k = = n l L n ( ) ( ) Boundary Conditions Y 0 , t = 0 = Y L , t

  29. n p æ ö ( ) ( ) ( ) 2 2 2 2 Prob x , t = Y x , t = A sin x cos w t è ø L h h nh p = = = , n = 1 , , ¥ K 2 L l 2 L n n n i . e . Momentum is Quantized n h 2 2 2 p æ h 2 ö 2 4 L 2 K = = = n = E è ø m 2 2 m 2 8 mL n n Kinetic energy = Total energy as electric é ù ê ú potential is zero inside box ë û Kinetic & Total energy are Quantized E = n E 2 n 1 2 h æ ö E = > 0 Zero Point Energy K è ø 2 8 mL 1 If electron drops from energy level b to energy level a the frequency of light emitted is h æ ö ( ) n = b - a Hz 2 2 è ø 2 ba 8 mL

  30. The Schrödinger Equation Schrödinger first guessed that the ( ) matter wave Y x , t would satisfy the linear wave equation ¶ Y w ¶ Y ¶ Y 2 2 2 2 = = v 2 2 2 2 2 ¶ t k ¶ x ¶ x just as string waves do, however this did not give the correct non relativistic energy for a free (traveling) electron p 2 i . e . E = = K 2 m nor did it give the correct spectrum for the hydrogen spectrum ( bound \ localized electron )

  31. The Equation for matter (in particular electrons ) that does give the correct energy is ( ) ( ) ¶ Y x , t ¶ Y x , t 2 h 2 i ( ) = - + U ( x ) Y x , t h ¶ t 2 m ¶ x 2 which is called Schrödinger's Time Dependent Wave Equation [ ] U ( x ) is the P . E . of the particle [ ] For free particles U ( x ) = 0 this equation has solutions of the form æ ö ( ) - i px Et ç ÷ ç ÷ ( ) x t ( ) = e - h Y = e , ikx i t è ø w [ ] E = w ; p = k h h

  32. æ æ ö ö è è ø ø æ ö ( ) - i px Et ç ÷ ç ÷ substituting = e into ( ) - h ( , ) w è ø ikx i t Y x t = e ( ) ( ) ¶ Y x , t ¶ Y x , t 2 h 2 i = - gives h ¶ t 2 m ¶ x 2 æ ö æ ö ( ) ( ) - - 2 i px Et i px Et iE ip ç ÷ ç ÷ h 2 ç ÷ ç ÷ - i e = - e h h h è ø è ø 2 m h h æ ö æ ö ( - ) ( - ) i px Et i px Et 2 p ç ÷ ç ÷ ç ÷ ç ÷ Û Ee = e h h è ø è ø 2 m p 2 Þ E = 2 m the non relativistic K. E . for a free particle

  33. ( ) ( ) ( ) Notice that Y x , t = e ( ) = F x W t - w ikx i t plugging this product into ( ) ( ) ¶ Y x , t ¶ Y x , t 2 h 2 ( ) = - + U ( x ) Y x , t gives ih ¶ t 2 m ¶ x 2 ( ) ( ) d W t ¶ F x é ù 2 h 2 ( ) ( ) ( ) x = - + U ( x ) F x W t ih F ê ú dt ë 2 m ¶ x û 2 ( ) ( ) 1 d W t 1 ¶ F x é ù 2 h 2 ( ) Û = - + ( x ) F x ih U ê ú ( ) ( ) W t dt F x ë 2 m ¶ x û 2 This equation can only have a solution if both sides are constant ( ) ( ) 1 d W t 1 ¶ F x é ù 2 h 2 ( ) & - + U ( x ) F x ih ê ú ( ) ( ) W t dt F x ë 2 m ¶ x û 2 E E

  34. ( ) 1 d W t = E ih ( ) W t dt ( ) d W t ( ) ih = E W t Þ dt iEt - ( ) Þ W t = Ce ; C is a constant h ( ) 1 h ¶ F x 2 2 é ù ( ) - + U ( x ) F x = E ê ú ( ) F x ë 2 m ¶ x û 2 2 h ¶ 2 é ù ( ) ( ) Þ - + U ( x ) F x = E F x ë 2 m ¶ x û 2 Time Independent Schrödinger Equation

  35. Solutions of differential equations are not completely characterized by the equation alone . The functions must also satisfy some boundary conditions, such as having a specified value at t = 0 . For the Schrödinger Equation the boundary conditions are ¥ ò ( ) 2 ( 1) Y x , t dx = 1 - ¥ ( ) (2 ) Y x , t is a continuous function in x ( ) d Y x , t (3 ) is a continuous function in x dx ( ) (4 ) Y x , t = 0 where U ( x ) = ¥

  36. Particle in a Box -- Again The potential energy of a particle in a box is ¥ for x < 0 ì ï [ ] U ( x ) = 0 for x Î 0 , L í ï ¥ for x > L î This gives the Time Independent Schroedinger Equation ¶ é 2 ù h 2 ( ) ( ) - + U ( x ) F x = E F x ë 2 m ¶ x û 2 2 æ h ö Which has solutions when E = n ; n = 1 , 2 , 2 K è ø 2 n 8 mL For n = 1 one can then solve the equation ¶ é ù æ h 2 h 2 2 ö ( ) ( ) - + U ( x ) F x = F x è ø 2 ë 2 m ¶ x û 1 8 mL 1 2 n p x æ ö ( ) Þ F x = A sin ; A is a constant that can be chosen to è ø 1 L ( ) normalize F x 1

  37. The Simple Harmonic Oscillator The potential energy of a particle that moves in SHM is 1 1 2 2 x 2 U ( x ) = k x = m w 2 2 k where w = & k is the force constant m This gives the Time Independent Schroedinger Equation ¶ 1 é ù 2 h 2 ( ) ( ) 2 2 - + m w x F x = E F x ë 2 m ¶ x 2 û 2 1 æ ö w Which has solutions when E = n + ; n = 0 , 1 , 2 , h K è ø 2 n For n = 0 one can then solve the equation w ¶ 1 é 2 ù h 2 h ( ) ( ) - + m w x F x = F x 2 2 ë 2 m ¶ x 2 û 2 2 æ ö w m - ç ÷ 2 ( ) x Þ F x = Be ; B is a constant that can be chosen to è 2h ø ( ) normalize F x

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