Lecture 16 Matter Waves & Wave Functions

LECTURE 16 MATTER WAVES & WAVE FUNCTIONS

Instructor: Kazumi Tolich Lecture 16

¨ Reading chapter 34-5 to 34-8 & 34-10 ¤ Electrons and matter waves n The de Broglie hypothesis n Electron interference and diffraction ¤ Wave-particle duality ¤ Standing waves and energy quantization ¤ Wave function ¤ Uncertainty principle ¤ A particle in a box n Standing wave functions de Broglie hypothesis

¨ In 1924 Louis de Broglie hypothesized: ¤ Since light exhibits particle-like properties and act as a photon, particles could exhibit wave-like properties and have a definite wavelength. ¨ The wavelength and frequency of matter: � = , and � = ¤ For macroscopic objects, de Broglie wavelength is too small to be observed. Example 1

¨ One of the smallest composite microscopic particles we could imagine using in an experiment would be a particle of smoke or soot. These are about 1 µm in diameter, barely at the resolution limit of most microscopes. A particle of this size with the density of carbon has a mass of about 10-18 kg. What is the de Broglie wavelength for such a particle, if it is moving slowly at 1 mm/s? Diffraction of matter

¨ In 1927, C. J. Davisson and L. H. Germer first observed the diffraction of electron waves using electrons scattered from a particular nickel crystal. ¨ G. P. Thomson (son of J. J. Thomson who discovered electrons) showed electron diffraction when the electrons pass through a thin metal foils.

¨ Diffraction has been seen for neutrons, hydrogen atoms, alpha particles, and complicated molecules.

¨ In all cases, the measured λ matched de Broglie’s prediction.

X-ray diffraction electron diffraction neutron diffraction Interference of matter

¨ If the wavelengths are made long enough (by using very slow moving particles), interference patters of particles can be observed. ¨ These figures show the build up of the electron two-slit interference pattern as the electrons arrive at the detector. Quiz: 1 Demo: 1

¨ Electron Diffraction ¤ Demonstration of circular diffraction patters from beam of electrons Quiz: 2 & 3

9 Wave-particle duality

¨ Light, normally thought of as a wave, exhibits particle properties when it interacts with matter. ¤ photoelectric effect ¤ Compton scattering ¨ Electrons, normally thought of as particles, exhibit the wave properties when they pass near the edges of obstacles. ¤ interference and diffraction ¨ All carriers of p and E exhibit both wave and particle characteristics. ¨ Everything propagates like a wave and exchanges energy like a particle. Standing waves and energy quantization

¨ Schrödinger (and others) developed quantum mechanics: ¤ Since particles have wavelike properties, they can produce standing particle waves. ¤ Since � = ℎ�, standing waves imply quantized energies. ¨ In quantum mechanics, a particle is described by a wave function � that obeys a wave equation called the Schrödinger equation. ¨ The form of the Schrödinger equation of a particular system depends on the potential energy functions associated with the forces acting on the particle. ℏ � − �Ψ �⃗, � + � �⃗ Ψ �⃗, � = �ℏ Ψ �⃗, � 2� �� You absolutely do not need to memorize the formula. Wave function

¨ The Schrödinger equation describes a single particle.

¨ The probability density � � , the probability per unit volume (or length in 1-D), of finding the particle as a function of position is given by � � = � � ¨ If we have a particle, the probability of finding the particle somewhere must be 1. Therefore the wave function must satisfy the normalization condition: � � �� = � � �� = 1 ¨ For � to satisfy the normalization condition, it must approach zero as � approaches infinity. Uncertainty principle

¨ If we use light with λ to measure the position of an object, x, its uncertainty, Δx, cannot be less than ~λ because of diffraction.

¨ If we use photons with pγ = h/λ to measure the momentum of an object, p, Δp of the object cannot be less than ~h/λ since the photon changes the momentum of the object upon scattering.

¨ The Heisenberg uncertainty principle states that:

¨ It is impossible to simultaneously measure both the position and the momentum of a particle with unlimited precision. ℏ ∆�∆� ≥ , where ℏ ≡ A particle in a box: conditions for �

¨ Consider a particle of mass m confined to a one-dimensional box of length �. ¤ Classical Mechanics: The particle with any values of energy and momentum bounces back and forth between the walls of the box. 2 ¤ Quantum Mechanics: The particle is described by a wave function �, and � describes the probability of finding the particle in some region. � 2 ¨ The particle is somewhere in the box:∫0 � � �� = 1 ¨ The particle is not outside the box: � � = 0 for � ≤ 0 and � ≥ �

¨ � is continuous everywhere: � 0 = 0 and � � = 0 ¤ This is the same boundary condition as the condition for standing waves on a string fixed at � = 0 and � = �. A particle in a box: standing wave function

¨ The normalization conditions yield the wave function. � = � ; � = 1,2,3, … � � = sin �� , where � = A particle in a box: allowed energies

¨ The standing wave condition yields the allowed energies.

¨ The lowest allowed energy, E1, is called ground state energy. ¨ The electron can make a transition to and from one energy state, Ei, to another, Ef, by the emission of a photon (if Ei > Ef). ¨ The frequency of the emitted photon is � = . ¨ The wavelength of the emitted photon is � = = . Quantum number

¨ The number “n” is called a quantum number.

¨ It characterizes ψn for a particular state and for the energy of that state, En. ¨ For a particle in a 1-D box, a quantum number arises from the boundary condition on � : � 0 = 0 and � � = 0. ¨ For a particle in a 3-D box, three quantum numbers arise, one associated with a boundary condition in each dimension. Quiz: 4 Large quantum number and classical results

¨ Probability: 2 2 ¤ The particle is most likely to be found near the maxima of ψn (x). The particle cannot be found where ψ = 0. ¤ For very large values of n, the maxima and minima are so closely spaced that ψ2 cannot be distinguished from its average value. The particle is equally likely to be found anywhere in the box, the same as in the classical result.

¨ Energies: ¤ The fractional energy difference of adjacent states becomes very small as the quantum number increases. ¤ For a very large n, energy quantization is not important.

¨ Bohr’s correspondence principle states: In the limit of very large quantum numbers, the classical calculation and the quantum calculation must yield the same results. Example 2

¨ An electron is in the initial state ni = 3 of an 1-D box of length 100 pm. If it is to make a quantum jump to the state nf = 6 by absorbing a photon, what must be the energy and wavelength of the photon? Harmonic oscillator potential well

¨ Consider a particle with mass, m, on a spring with force constant, k. ¨ Potential energy function for a harmonic oscillator is parabolic: � � = �� = ��, where � = is the natural frequency of the oscillator.

¨ Classically, the object oscillates between ±A, and its total energy, E, which can have any nonnegative value, including zero: � = ��.

¨ Normalizable ψn(x) occur only for discrete values of the energy En given by Note that the ground state energy is not 0.

Equally spaced levels: hf0 Example 3

¨ An electron in a harmonic oscillator is initially in the n = 4 state. It drops to n = 2 state and emits a photon with wavelength 500 nm. What is the ground state energy of this harmonic oscillator?