School of Mathematical and Physical Sciences PHYS August, PHYS1220 – Quantum Mechanics Lecture 4 August 27, 2002 Dr J. Quinton Office: PG 9 ph

School of Mathematical and Physical Sciences PHYS August, The Correspondence Principle Any theory must match the well-known laws of classical physics if the conditions match the classical case. This is known as The Correspondence Principle Recall from special relativity that when v<<c, the theory must simplify to Newtonian physics eg relativistic kinetic energy becomes if you take the expansion and make v<<c In Quantum Mechanics, the same applies in going from microscopic to macroscopic situations (ie when the system >> de Broglie l ) As the quantum number, n, approaches infinity, any real system should behave in a way that is consistent with classical physics Eg in the Bohr model, the discrete energy levels E n get closer and closer together and as n → , they essentially become ‘continuous’ The same applies for r n and L as n → . The exercise is left to the student

School of Mathematical and Physical Sciences PHYS August, Quantum Mechanics Bohr’s model contained a remarkable mixture of classical and quantum concepts, thus it provoked much thought about the wave nature of matter, light and the laws of how they interact with one another This led to the development of a comprehensive theory to describe microscopic phenomena, started independently in (1925) by Werner Heisenberg (Matrix mechanics, Nobel Prize 1932)  Heisenberg’s approach employs matrices and matrix functions and although very powerful, is mathematically complicated and less suitable for teaching elementary concepts. Erwin Schrödinger (Wave mechanics, Nobel Prize 1933)  Schrödinger’s approach uses multivariable functions and operators. Their approaches were very different from one another but their theories are fully compatible

School of Mathematical and Physical Sciences PHYS August, The Uncertainty Principle Every measurement has an associated uncertainty According to classical physics, there is no limit to the ultimate refinement of the apparatus or measuring procedure It is possible to determine everything to infinite precision Quantum Theory predicts otherwise. With his matrix mechanics, Heisenberg showed the existence of what is called the Heisenberg Uncertainty Principle If a measurement of position is made with precision  x and a simultaneous measurement of momentum component p x is made with precision  p x, then It is fundamentally impossible to simultaneously measure the exact position and exact momentum of a particle Other uncertainty relations are

School of Mathematical and Physical Sciences PHYS August, Uncertainty Principle Example A ‘thought’ experiment of Heisenberg’s Suppose you want to simultaneously measure the position and momentum of an electron as precisely as possible with a powerful light microscope In order to determine the electron’s location (ie making  x small ~ l ) at least one photon of light (with momentum h/ l must be scattered (as in (a)) But the photon imparts an unknown amount of its momentum to the electron (as in (b)), thus altering it’s path and speed! ie  p ~ h/ l becomes larger! The very light that allows you to determine the position changes the momentum by some undeterminable amount In making measurements on microscopic scales, you must now appreciate that you cannot make a measurement without interacting with the very thing that you are attempting to measure! So  x  p ~ h

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School of Mathematical and Physical Sciences PHYS August, Is the Bohr Model Realistic? Question: According to the Bohr model, the electron in the ground state moves in a circular orbit with the Bohr radius r 1 =0.529x m, at a speed of 2.2x10 6 ms -1 (check it for yourself!). In view of the HUP, is the model realistic? Answer: Because the model assumes that the electron is located at r 1, the uncertainty  r is zero. If the magnitude of the total momentum of the electron is mv, then the radial component of momentum must be less than or equal to this value According to the uncertainty principle, the minimum uncertainty in the radial position is therefore which is ~ r 1 ! so the Bohr model is not realistic

School of Mathematical and Physical Sciences PHYS August, Philosophical Implications Newtonian physics is completely deterministic. If you know the configuration of a system at any point in time then you can predict its future and also extrapolate its past Quantum mechanics (QM) has drastically altered our viewpoint Because the wave nature dominates on atomic scales, we must relinquish determinism and accept a probabilistic approach.  The expected position of a microscopic particle (such as an electron that is moving around a nucleus) can only be predicted by calculating a probability, which in turn indicates an expected statistical average over many measurements. Even macroscopic objects that are made up of many atoms are governed by probability rather than strict determinism.  eg QM predicts a finite (though negligibly small) probability that an thrown object (comprising many atoms) will suddenly curve upward rather than follow a parabolic trajectory However when large numbers of objects are present in a statistical situation, deviations from the most probable approach zero, and thus obey classical laws with very high probability, giving rise to an apparent ‘determinism’ Many people opposed this but ultimately had to accept it. At the time Einstein believed that “God does not play dice with the universe”

School of Mathematical and Physical Sciences PHYS August, The Wave Function and its Interpretation In Quantum Mechanics, each object (particle or the system itself) is represented by a ‘matter wave’ and is described by a (unique) wave function,  (x,y,z,t). The wavelength is established by de Broglie, but what is the physical meaning of the amplitude? One way to interpret the wave function is that it plays the same role that the electric field vector plays in the wave theory of light Recall that Intensity  (Amplitude) 2 In a similar way,  2 represents an ‘intensity’ or alternatively, the probability of detecting the object with wave function  The magnitude of the wave function  (itself generally a complex quantity) may vary in x,y,z or t, but the probability of detecting the particle will be greater where and/or when the amplitude is large. If  represents a single electron (say in an atom) then the value of |  | 2 dV at a certain point in space and time represents the probability of finding the electron within the volume dV about the given position at that time – Max Born, 1928 (Nobel Prize 1954)

School of Mathematical and Physical Sciences PHYS August, Properties of Wave Functions Wave functions of particles must possess certain properties to be useful quantum mechanically. The function must be continuous The function must be differentiable the particle exists and so the the probability of finding it throughout all of space must be equal to 1. When this is the case, the function is said to be ‘normalised.’ A function must be normalised for the probability to make sense. eg the probability of detecting an electron with a wave function y between x=a and x=b is determined by Expectation values,. The expectation value of any quantity is the statistical average after many measurements of the quantity are made. For example, the expectation value of position is given by

School of Mathematical and Physical Sciences PHYS August, Eigenvalues and Eigenfunctions Consider In this example, the derivative is an operator on the function f(x). Because the function f(x) is returned (multiplied by a constant) after it is acted on by the derivative operator, f(x) is said to be an eigenfunction of the derivative operator Or more specifically in this case, the exponential function is an eigenfunction of the derivative operator The constant that is returned as a multiplier of an eigenfunction is called its eigenvalue Here, the constant k is an eigenvalue of the eigenfunction f (x). Question: is an eigenvector of the 2 nd derivative?

School of Mathematical and Physical Sciences PHYS August, Recall from wave theory that any travelling wave can be represented mathematically by where k is the angular wavenumber, w the angular frequency, A is the amplitude and f 0 is an initial phase Therefore both momentum and energy are contained in the terms describing the wave Waves Revisited Now, noticing that and then the wave can be expressed by

School of Mathematical and Physical Sciences PHYS August, The Schrödinger Equation The Schrödinger equation cannot be derived from first principles. It appears as a postulate, just as Newton’s second law does We will consider only one dimensional, steady state problems (where  and the potential U are only a function of spatial x and independent of time). The 1-D Time-Independent Schrödinger Equation (T.I.S.E.) is where m is the mass of the particle The equation is essentially the total energy of the particle

School of Mathematical and Physical Sciences PHYS August, The Schrödinger Equation II The T.I.S.E. is based upon conservation of energy, so all 1-D time-independent systems must obey it Note that the T.I.S.E. is an operator equation, however. The wave function of any real object must be an eigenfunction of Schrödinger’s equation, with its corresponding eigenvalue equal to the object’s energy, E. In other words, once you know the eigenfunction of a particle (or its state) you can just substitute it into Schrödinger’s equation to calculate the energy Well that is quite a bit of theory, now let’s use it for some simple situations

School of Mathematical and Physical Sciences PHYS August, Free Particles The simplest wave function to describe is that of a free particle. It does not have any potential acting on it and therefore no forces. The T.I.S.E. is therefore Which can be written This is the second order differential equation for a harmonic oscillator with general solution Note that k can have any value (ie the energy can be chosen from a continuous range). Note also that  p x is zero, so  x → 

School of Mathematical and Physical Sciences PHYS August, Free Particles II Given the wave function, let’s substitute it back into the T.I.S.E. So If we had guessed the wave function, we could have computed the energy of the free particle (and how it depends upon k) That is a simple example of the power of Schrödinger’s equation

School of Mathematical and Physical Sciences PHYS August, Wave Packets To represent a particle that is well localised (ie its position is known to be within a small region of space), we use the concept of a wave-packet To describe this requires a wave function that is the sum of many sinusoidal plane waves of slightly different wavelengths (cf beats). The smaller the value of  x, the more terms are needed in the sum. Because each term in the sum has a unique wavelength (and therefore momentum), the sum does not have a definite momentum. Rather, it has a range of momenta, so  p x is non-zero