2.2 Schrödinger s wave equation

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1 2.2 Schrödinger s wave equation Slides: Video Schrödinger wave equation introduction Text reference: Quantum Mechanics for Scientists and Engineers Section Chapter 2 introduction

2 Schrödinger s wave equation Quantum mechanics for scientists and engineers David Miller

4 2.2 Schrödinger s wave equation Slides: Video From de Broglie to Schrödinger Text reference: Quantum Mechanics for Scientists and Engineers Sections

5 Schrödinger s wave equation From de Broglie to Schrödinger Quantum mechanics for scientists and engineers David Miller

6 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 34 h J s Now we want to use this to help construct a wave equation

7 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 2 d 2 2 k 2 with k 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))

8 A Helmholtz wave equation In three dimensions, we can write this as x y z k 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

9 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 / so k p / Hence we can rewrite our Helmholtz equation 2 2 p 2 or p

10 From Helmholtz to Schrödinger If we are thinking of an electron, we can divide both sides by its mass m o to obtain p 2mo 2mo But we know from classical mechanics that 2 p kinetic energy of electron 2m o and in general Total energy ( E)=Kinetic energy + Potential energy ( V r )

From Helmholtz to Schrödinger So Kinetic energy = p 2 /2m o 2 2 2 p Hence our Helmholtz equation 2mo 2mo 2 2

11 From Helmholtz to Schrödinger So Kinetic energy = p 2 /2m o p Hence our Helmholtz equation 2mo 2mo 2 2 becomes the Schrödinger equation EV r 2m o = Total energy ( E) - Potential energy ( V r ) or equivalently m 2 2 Vr E 2 o

12 Schrödinger s time-independent equation We can postulate a Schrödinger equation for any particle of mass m 2m 2 2 V r E Formally, this is the time-independent Schrödinger equation

13 Probability densities Born s postulate is that the probability Pr of finding an electron near any specific point r in space is proportional to the modulus squared of the wave amplitude r r 2 can therefore be viewed as a probability density with r called a probability amplitude or a quantum mechanical amplitude r 2

15 2.2 Schrödinger s wave equation Slides: Video Diffraction by two slits Text reference: Quantum Mechanics for Scientists and Engineers Section 2.3 (first part)

16 Schrödinger s wave equation Diffraction by two slits Quantum mechanics for scientists and engineers David Miller

17 Young s slits An opaque mask has two slits cut in it, a distance s apart s

18 Young s slits We shine a plane wave on the mask from the left s

19 Young s slits What will be the pattern on a screen at a large distance z o? s? z o

20 Young s slits The slits as point sources give an interference pattern

21 Young s slits The distance from the upper source to point x on the screen is 2 2 x s/2 zo x s /2 x s 2 z o o o o x s/2 z z 1 x s/2 / z z x s/2 /2z o z x /2 z s /8 z sx/2z 2 2 o o o o o

22 Young s slits s 2 The distance from the lower source to point x on the screen is z xs/2 /2z o o x s/2 zo z x /2 z s /8 z sx/2z 2 2 o o o o x s /2 x z o

23 Young s slits For large z o the waves are approximately uniformly bright i.e., using exponential waves for convenience s x exp ik x s / 2 z o exp ik x s / 2 z o Using our approximate formulas for the distances gives s x exp i exp ik sx/ 2zo exp ik sx/ 2zo 2 2 where k z x /2 z s /8z o o o

24 Young s slits Now so i i exp exp 2cos sx sx s x expi exp ik exp ik 2zo 2zo sx sx expicosk expicos 2z o z o so the intensity of the beam 2 x cos sx / z 1 cos2 sx / z s o o

25 Young s slits The interference fringes are spaced by d s zo / s ds s zo

26 Young s slits This allows us to measure small wavelengths d s s / zo ds s zo

28 2.2 Schrödinger s wave equation Slides: Video Interpreting diffraction by two slits Text reference: Quantum Mechanics for Scientists and Engineers Section 2.3 (second part)

29 Schrödinger s wave equation Interpreting diffraction by two slits Quantum mechanics for scientists and engineers David Miller

30 Young s slits If the upper slit is blocked no interference pattern

31 Young s slits If the lower slit is blocked no interference pattern

Quantum Mechanics for Scientists and Engineers. David Miller

Quantum Mechanics for Scientists and Engineers David Miller The particle in a box The particle in a box Linearity and normalization Linearity and Schrödinger s equation We see that Schrödinger s equation