Quantum Mechanics - Modern Physics - Lecture Slides, Slides of Physics

Download Quantum Mechanics - Modern Physics - Lecture Slides and more Slides Physics in PDF only on Docsity!  6.1 The Schrödinger Wave Equation  6.2 Expectation Values  6.3 Infinite Square-Well Potential  6.4 Finite Square-Well Potential  6.5 Three-Dimensional Infinite-Potential Well  6.6 Simple Harmonic Oscillator  6.7 Barriers and Tunneling Quantum Mechanics I think it is safe to say that no one understands quantum mechanics. Do not keep saying to yourself, if you can possibly avoid it, “But how can it be like that?” because you will get “down the drain” into a blind alley from which nobody has yet escaped. Nobody knows how it can be like that. - Richard Feynman docsity.com 6.1: The Schrödinger Wave Equation  The Schrödinger wave equation in its time-dependent form for a particle of energy E moving in a potential V in one dimension is  The extension into three dimensions is where is an imaginary number. docsity.com Properties of Valid Wave Functions Boundary conditions 1) In order to avoid infinite probabilities, the wave function must be finite everywhere. 2) In order to avoid multiple values of the probability, the wave function must be single valued. 3) For finite potentials, the wave function and its derivative must be continuous. This is required because the second-order derivative term in the wave equation must be single valued. (There are exceptions to this rule when V is infinite.) 4) In order to normalize the wave functions, they must approach zero as x approaches infinity.  Solutions that do not satisfy these properties do not generally correspond to physically realizable circumstances. docsity.com Time-Independent Schrödinger Wave Equation  The potential in many cases will not depend explicitly on time.  The dependence on time and position can then be separated in the Schrödinger wave equation. Let , which yields: Now divide by the wave function:  The left side of Equation (6.10) depends only on time, and the right side depends only on spatial coordinates. Hence each side must be equal to a constant. The time dependent side is docsity.com  We integrate both sides and find: where C is an integration constant that we may choose to be 0. Therefore This determines f to be  This is known as the time-independent Schrödinger wave equation, and it is a fundamental equation in quantum mechanics. Time-Independent Schrödinger Wave Equation Continued docsity.com  The position x is its own operator as seen above.  The time derivative of the free-particle wave function is Substituting ω = E / ħ yields  The energy operator is  The expectation value of the energy is Position and Energy Operators docsity.com Comparison of Classical and Quantum Mechanics  Newton’s second law and Schrödinger’s wave equation are both differential equations.  Newton’s second law can be derived from the Schrödinger wave equation, so the latter is the more fundamental.  Classical mechanics only appears to be more precise because it deals with macroscopic phenomena. The underlying uncertainties in macroscopic measurements are just too small to be significant. docsity.com 6.2: Expectation Values  The expectation value is the expected result of the average of many measurements of a given quantity. The expectation value of x is denoted by <x>  Any measurable quantity for which we can calculate the expectation value is called a physical observable. The expectation values of physical observables (for example, position, linear momentum, angular momentum, and energy) must be real, because the experimental results of measurements are real.  The average value of x is docsity.com Some expectation values are sharp some others fuzzy, continued I For any observable, fuzzy or not If not fuzzy, ΔQ = 0 Because <Q2>= <Q>2 x may as well stand for any kind of operator Q docsity.com Some expectation values are sharp some others fuzzy, continued II  Eigenvalues of operators are always sharp (an actual – physical - measurement may give some variation in the result, but the calculation gives zero fuzziness  Say Q is the Hamiltonian operator A wavefunction that solves this equation is an eigenfunction of this operator, E is the corresponding eigenvalue, apply this operator twice and you get E2 – which sure is the same as squaring to result of applying it once (E) docsity.com 6.3: Infinite Square-Well Potential  The simplest such system is that of a particle trapped in a box with infinitely hard walls that the particle cannot penetrate. This potential is called an infinite square well and is given by  Clearly the wave function must be zero where the potential is infinite.  Where the potential is zero inside the box, the Schrödinger wave equation becomes where .  The general solution is . docsity.com 6.4: Finite Square-Well Potential  The finite square-well potential is  The Schrödinger equation outside the finite well in regions I and III is or using yields . Considering that the wave function must be zero at infinity, the solutions for this equation are docsity.com  Inside the square well, where the potential V is zero, the wave equation becomes where  Instead of a sinusoidal solution we have  The boundary conditions require that and the wave function must be smooth where the regions meet.  Note that the wave function is nonzero outside of the box. Finite Square-Well Solution docsity.com Penetration Depth  The penetration depth is the distance outside the potential well where the probability significantly decreases. It is given by  It should not be surprising to find that the penetration distance that violates classical physics is proportional to Planck’s constant. docsity.com 6.6: Simple Harmonic Oscillator  Simple harmonic oscillators describe many physical situations: springs, diatomic molecules and atomic lattices.  Consider the Taylor expansion of a potential function: Redefining the minimum potential and the zero potential, we have Substituting this into the wave equation: Let and which yields . docsity.com Parabolic Potential Well  If the lowest energy level is zero, this violates the uncertainty principle.  The wave function solutions are where Hn(x) are Hermite polynomials of order n.  In contrast to the particle in a box, where the oscillatory wave function is a sinusoidal curve, in this case the oscillatory behavior is due to the polynomial, which dominates at small x. The exponential tail is provided by the Gaussian function, which dominates at large x. docsity.com Analysis of the Parabolic Potential Well  The energy levels are given by  The zero point energy is called the Heisenberg limit:  Classically, the probability of finding the mass is greatest at the ends of motion and smallest at the center (that is, proportional to the amount of time the mass spends at each position).  Contrary to the classical one, the largest probability for this lowest energy state is for the particle to be at the center. docsity.com Probability of Reflection and Transmission  The probability of the particles being reflected R or transmitted T is:  The maximum kinetic energy of the photoelectrons depends on the value of the light frequency f and not on the intensity.  Because the particles must be either reflected or transmitted we have: R + T = 1.  By applying the boundary conditions x → ±∞, x = 0, and x = L, we arrive at the transmission probability:  Notice that there is a situation in which the transmission probability is 1. docsity.com Tunneling  Now we consider the situation where classically the particle does not have enough energy to surmount the potential barrier, E < V0.  The quantum mechanical result, however, is one of the most remarkable features of modern physics, and there is ample experimental proof of its existence. There is a small, but finite, probability that the particle can penetrate the barrier and even emerge on the other side.  The wave function in region II becomes  The transmission probability that describes the phenomenon of tunneling is docsity.com Uncertainty Explanation  Consider when κL >> 1 then the transmission probability becomes:  This violation allowed by the uncertainty principle is equal to the negative kinetic energy required! The particle is allowed by quantum mechanics and the uncertainty principle to penetrate into a classically forbidden region. The minimum such kinetic energy is: docsity.com