The Experimental Observations Associated with Photoelectric Effect - Assignment Example

Quantum Mechanics Name: Course: Professor: Institution: City & State: Date: Quantum Mechanics I 2009 1. a. The experimental observations associated with photoelectric effect i. It’s clearly observed that for a given metal and frequency of the incident radiation, the rate of photoelectric emission is directly proportional to the intensity of the incident light. Also, for a given metal of a particular work function, an increase in intensity of the incident beam of light increases the magnitude of the photoelectric current while the stopping voltage remains the same. For a given metal, the threshold frequency exists for the incident radiation below which no photoelectrons can be emitted. If the incident light is linearly polarized, the direction of distribution of the emitted electrons peaks in the direction of the field. ii. If an electromagnetic radiation of very short wavelength is emitted and collides with matter, energy is absorbed by the matter (liquids, gas, and solids) and as a result of this; electrons are released from the matter. Photoelectric effect follows the rule that any given metal has a threshold frequency f0 given by f0 =  Where:  = minimum energy h = Planck’s constant Below this frequency, the light emitted will not bring about the photoelectric effect. The diagram below show the light matter interaction b. i. The de Broglie relations state that the wavelength is inversely proportional to the momentum of a particle and that the frequency is directly proportional to the particle’s energy. ii.  =  =  Where:  = Anguler wave number  = wavelength f= frequency v = speed c. For a classical free particle: The momentum is given by: P = mv And the energy is given by: E = ½ mv2 Where: m = mass of the particle v= velocity of the particle Therefore the Schrödinger’s equation for a free particle is:  =  For a particular momentum, the plane is given as: With constant: Where: r is the position vector t is time k is wave vector And 𝝎 is the angular velocity The wave function must be normalized since the integral of * over all the space must be united. Therefore, momentum P is And the expected energy is given by: Solving for k and 𝛚 and then substituting them to the equation above:  And the group velocity is given as: Therefore, it is clear that a free particle wave function may be represented as: Where the integrals are in the space of k Therefore  d. 1. Quantum tunneling is the quantum mechanical phenomenon where the particle tunnels through a barrier that classically cannot surmount. 2. The alpha particle inside the atomic nucleus The figure below shows the behavior of an alpha particle inside the atomic nucleus. When inside the atomic nucleus, the alpha particle cannot escape and therefore alpha decay cannot occur since an alpha particle within the nucleus is deemed to have little total energy compared to the potential barrier of the nucleus. Alpha particle outside the nucleus e. Physical interpretation of a wave function The born interpretation of the wave function is concerned with the likelihood of finding of particle at dissimilar positions in the space. This can be seen in case for a wave function of a particle with the value x at a given Y. Therefore, the probability of finding the particle between Y and Y + dY is proportional to |x|2 Where x is the probability amplitude and |x|2 is the probability density. From the wave function given therefore, it is easier to get the information about the particle where the kinetic energy can be found in the wave function. Stationary state wave functions: These are called stationary since the particles are in a stationary state and they are deemed to stay in that state over an observable time. At this state, the particle is said to have constant probability distribution in terms of its velocity, position and the particle’s spin. This applies only if the rest of the particle’s system is static. The expected value of energy for a given particle in the first energy state is given as:  And for the second state:  Section B Question 2 i. Compton experiment- this is a type of scattering whereby the x-rays undergo inelastic scattering of photons in matter. This inelastic scattering results in the X-ray photon energy decrease and as a result the wavelength increases. During this experiment the x-ray transfers some of its energy to a scattering electron thus ionizing the electron. The rest of the energy is scattered with the photon. The diagram below shows the Compton scattering of the x-ray of the metal target. ii. According to Planck’s relationship the scattered photon has lower energy therefore having a long wavelength. Assuming the nature of light and applying the law of conservation of energy and conservation of momentum, this explains the collision of electron and the photon. According to the Compton formula the wavelength shift increases with the scattering angle. Therefore, the scattering of the x-rays with the longer wavelength are scattered from electrons in a carbon target. The Compton formula: iii. For a Gaussian propagating in a free space is given by: According to the formula Where: l = the wavelength (m) W= the fridge spacing (m) S = the slit spacing (m) D = the distance from the slits to the screen (m)  = wavelength E = hv Where: E = energy h=planck’s constant v = velocity v= E/h 𝜆 = hf Where:  the integer of the number of slits E = energy h = Planck’s constant Therefore: wavelength l = hf Therefore: Question 3 a. The uncertainty principle states that the precise inequalities that constrain certain pairs of physical properties, such as measuring the present position while determining future momentum of a particle. b. For the summation of a special wave packet by summation of two plane waves. Where: c = speed of the wave’s propagation in a given medium Using  the wave equation has plane wave solutions In which, the first term in the equation represents a wave propagating in the x direction A general form of a wave packet is represented as: The factor  comes from the Fourier transform conventions  Have the linear coefficients of the linear superposition Quantum Mechanics I 2010 (a) (i) Principle features and experimental observations associated with photoelectric effect. It’s clearly observed that for a given metal and frequency of the incident radiation, the rate of photoelectric emission is directly proportional to the intensity of the incident light. Also, for a given metal of particular work function, an increase in intensity of the incident beam of light increases the magnitude of the photoelectric current while the stopping voltage remains the same. For a given metal, the threshold frequency exists for the incident radiation below which no photoelectrons can be emitted. (ii) Compton scattering experiment During Compton scattering, the light photons collide with the electrons thus decreasing the energy which in turn increases the wavelength of X-ray or gamma ray. This experiment is an example of an inelastic scattering and involves inelastic collisions between the photons and the electrons. Principle experimental observations of the Compton scattering experiment are that, during the experiment, the electrons and the energy photons interact and as a result, the electrons given or released ensure conservation of the momentum. Also, if the photon has lower energy, it can still eject an electron from the host atom. (iii) Both (i) and (ii) have the actions of the photons in that, when they collide, metal electrons are released as packets or particles. The electrons that are released have some energy in them and therefore they move as packets of energy and have particle-like properties of electro-magtic radiation. (iv) Why free electrons cannot absorb a photon To conserve the principle of energy conservation, the electrons cannot absorb photons and therefore, only electrons bound can absorb photons. If the incident light is linearly polarized, the direction of distribution of emitted electrons peaks in the direction of the field (b) i. Conventional interpretation of the wave function as stated by Max Born (the Copenhagen interpretation). The Max Burn interpretation tries to explain the results of experiments and their related mathematical formulations. The act of measurement causes the probabilities of calculated measurement values. ii. The probability distribution function for the wave function If 𝛙(x) represents a simple one dimensional single-particle and therefore, the probability that the particle is located at distant x can be given as: Since this equation is mathematically constant, therefore it is in a stationary state. (c) (iv) Collapse of a wave function following a quantum measurement. This is a situation whereby the wave function is in a superposition of several different possible eigenstates. This superposition appears to reduce to a single one and this reduces the possibility of the physical possibilities. Wave function collapse is one of the processes through which quantum systems evolve in time and this is according to the law of quantum mechanics. Section B Question 3 a. The Heisenberg uncertainty principle states precise inequalities that constrain certain pairs of physical properties, such as measuring the present position while determining future momentum of a particle (Mehra & Rechenberg, 2001, p. 23). b. For the summation of a special wave packet by summation of two plane waves; Where: c = speed of the wave’s propagation in a given medium Using  the wave equation has plane wave solutions Where. The first term in the equation represents a wave propagating in the x direction A general form of a wave packet is represented as: The factor  comes from the Fourier transform conventions  Have the linear coefficients of the linear superposition How to construct a wave packet representing an isolated single particle? This uses probability amplitude in quantum mechanics to describe the quantum state of a particle in a system. Using a function of space or momentum if possible, that of time which returns the probability amplitude of momentum or of position for a particle of subatomic nature helps to explain the wave packet of an isolated single particle. Also, using the equation of function from space, it maps the possible states of the system in the complex numbers that define the laws of quantum mechanics to give the wave function. This definition gives the wave-particle duality of position and time. Question 4 The time-dependent Schrodinger equation for a particle of mass m, in a potential ; For a general quantum system;  For a single isolated particle: Where:  is the kinetic energy operator and m is the particle’s mass  = the Laplace operator = for a Cartesian coordinates  V(x) = time dependent potential energy at x  is the wave function for a particle at x position at time t For the time dependent function;  Thus the equation describes a standing wave. Thus, Eigen functions of the Hamiltonian operator For the differential equation:   For a= 10-10 m C =  = 8.319 m E = hf References Mehra, J. & Rechenberg, H., 2001. The historical development of Quantum Theory. New York: Springer. Read More