3. Quantum Mechanics in 3D

Transcription

1 3. Quantum Mechanics in 3D 3.1 Introduction Last time, we derived the time dependent Schrödinger equation, starting from three basic postulates: 1) The time evolution of a state can be expressed as a unitary linear transformation of the state, represented by the time-evolution operator expressed in terms of the time independent hamiltonian. 2) We required that the transformation or evolution of the state conserve probability (i.e. the number of possible states a system may find itself in is the same before and after the time evolution), and 3) we established that this form of time evolution is valid for time independent central forces in closed systems. The hamiltonian was then identified as the generator of displacements of wave functions, or states, in time, being equal to the application of the time derivative of the state or function. This was a direct result of investigating the change of the time evolution operator as a function of time (i.e. taking the time derivative of the evolution operator). Therefore, the use of the Schrödinger equation is strictly valid only under the above named conditions. All other cases may be solved by introducing changes from the above conditions as a perturbation. The canonical example, at the subatomic level, of a closed system with central time independent forces is the hydrogen atom. Before we proceed to solve the Schrödinger equation in a form describing the hydrogen atom, we must deal with a few prerequisites, including the study of general solutions to the 3D Schrödinger equation, the study of angular momentum and the addition of angular momentum, as well as the solution for the two-body problem (rather than a single object in a potential ). As we will see, these subjects are all closely related and, in the order they are stated, will lead to a complete quantum mechanical description of the hydrogen atom. Along with the solution of the harmonic oscillator problem, this represents the single most important result to come out of quantum mechanics. To give you a sneak-peek of why the above named subjects are all related, consider the following: The total angular momentum of a single particle is given by its spin (its rotation about an axis which is fixed with respect to its body, preferably with the origin at its center of mass; i.e. a body centered coordinate system ) and its orbital angular momentum with respect to the origin of some fixed coordinate system: As we will see later, the total angular momentum Ĵ 1 is conserved for central forces 1) (The bold font indicates that this is a vector in three dimensional Euclidean space, and the hat indicates that this is an operator, so Ĵ = Ĵ x + Ĵ y + Ĵ z, where each x,y,z component is an n-dimensional matrix in angular momentum space: For example, a spin ½ particle may have two angular momentum eigenvalues (±½). Therefore, the angular momentum space and the matrices representing the x,y,z operators have dimension 2).

2 (and most others) and, if given for a number of particles, can be added to give the new total angular momentum of the system. In fact, if you know the spin S 1 of a particle and the orbital angular momentum L 1 with respect to this particle, of a second particle which is passing by, then you can add the two, to obtain the total angular momentum of the system. One may be able to imagine a number of situations in which this may be useful, and to put this idea into some context, the simplest thing to do is to apply this two-body situation to the hydrogen atom. When doing so, we will find that both the proton and the electron have intrinsic spin (i.e. body centered rotation) and that they have a relative orbital angular momentum about each other an that this, together with the radial confinement of the electron (the fact that it can not be removed from the proton farther than its energy allows it to), will provide the complete and very exact description of its energy spectra. The atom itself may of course be moving and may, therefore, have its own orbital angular momentum with respect to some point. However, for the purpose of solving the Schrödinger equation for the hydrogen atom, which we consider to be a closed system, the motion of its center of mass about this point is unimportant. Note however, that the motion of the atom in an external magnetic or electric field violates this view and therefore violates all of the postulates (see previous page) based upon which the Schrödinger equation was derived. However, as long as these external effects are small and/or happen over short periods of time, they can be treated in perturbation theory, as we will see later on. I keep referring to the total, orbital, and spin angular momenta (J,L,S) as though they represent the actual values of what is measured or calculated. As indicated in the foot notes, they represent vectors, like they do in classical mechanics. However, as we will find out later, in quantum mechanics neither do they add like normal vectors, nor do we know their exact orientation. All we know is their magnitude and their projection onto some axis. So we can know but nothing more. We will find that knowing any one of the components J i (i = x,y,z) still allows us to know J 2, but not any of the other two components. Here, J represents all forms of angular momentum, including spin and orbital angular momenta. 1) The bold font indicates that this is a vector in three dimensional Euclidian space, corresponding to the eigenvalues of the three components of the vector angular momentum operator (see previous page foot note and note the absence of the hat notation here). Thus, the components of these vector are numbers, while the components of the vectors with hats (previous page) are matrices.

3 We will also discover that, contrary to the classical case, where the angular momentum of a system of particles L = Iω (I is the inertia tensor) is allowed to take on any value, in quantum mechanics or, in general, nature at the microscopic level, the eigenvalues of angular momentum can only take on discrete values, separated by equal intervals. This is referred to as space quantization. So how can we, with these limitations, determine the state of even a simple system? And why do we have the discrete values? Well, the answer is that 1) we don t actually care what the orientation of J,L, or S is, because the mutual exclusion of their components represents our ignorance of the exact location of a particle, which is a direct result of the uncertainty relations and the probabilistic nature of quantum mechanics. In other words, quantum mechanics is not equipped to tell us anything more and experimental verification has led us to the conclusion that we can t know anything more about microscopic systems. 2) A change in angular momentum corresponds to a change in energy of the system which can and does only happen when photons 1 are created or annihilated (emitted or absorbed), so that angular momentum changes have to occur in discrete steps. We will start by investigating the three dimensional form of the Schrödinger equation and obtain the angular commutation relations explicitly, again, as we have done last time for position and momentum commutation, by correspondence with the classical rules, this time for the angular momentum equation of motion. We will calculate the so called angular momentum selection rules and solve the angular momentum eigenvalue problem, based on the abstract methods developed in the second part of lectures. We will find explicit angular solutions of the Schrödinger equation in the spherical coordinate representation, obtaining explicit eigenfunctions in the coordinate representations. 3.1 The general three dimensional Hamiltonian The time-independent Schrödinger equation in one dimension is given by If we want to extend this to three dimensions, we need to come to the conclusion that one can simultaneously measure the position of a particle in all three dimensions. Otherwise, every three dimensional problem would reduce to a one dimensional problem by default. This is something that is almost innate to us, since we are so used to writing down 1) In saying this, I am staying within the confines of electromagnetic interactions.

4 diagonal three dimensional matrices in the usual Cartesian coordinate representation, but it isn t obvious at all that this is true in QM. Try and imagine the consequences if one couldn t measure the position of a particle in all three dimensions simultaneously? If we are willing to accept simultaneous position measurability in all directions (this shouldn t really cost you any sleep), then we can write down the commutation relations as follows, According to which we can write simultaneous eigenstates and use the notation and define our wave functions in correspondence to the notation developed earlier. Then, by construction: Based on our earlier derivation of the momentum operator in one dimension, the momentum operators corresponding to the three coordinate components are The properties of partial differentiation can be used to show that And the commutation relation between position and momentum can be shown to satisfy (proof in class) : Together, these spatial and momentum commutation relations are referred to as the canonical commutation relations. We can then write down the hamiltonian for a particle of mass m moving in three dimensions, as a generalization of the earlier stated one dimensional case: which, when inserted into the Schrödinger equation, produces

5 3.2 Separation of Variables in Cartesian coordinates If the potential term happens to separate into a sum over three terms each involving only one of the spatial dimensions (examples include the particle in a finite box of side length L or the 3D harmonic oscillator): Then the resulting differential equation can be satisfied by wave functions that are factorized into individual functions of x,y,z : And we obtain three separate one dimensional equations: And the total energy is then given by: We will show this explicitly in class. We shall return to this situation when we talk about many particle system. However, the more relevant situation for now is obtained when the potential turn out to be a so called central or spherically symmetric potential.

6 3.3 The Hamiltonian with a spherically symmetric potential If the potential experienced by a particle in three dimensions is spherically symmetric then the natural coordinates for the solution of the Schrödinger equation are spherical coordinates. The hamiltonian for such a potential takes on the form We will follow mostly Stephen Gasiorowicz in his derivation of the Schrödinger equation in spherical coordinates : This emergence of this particular form is bit mysterious when simply written down like this, so we will invest some time to derive in class (either this time or next time). The general solution can be separated into a radial and angular part: Which will be derived explicitly in class as well. Our aim over the next 3 weeks will be to solve for both the angular and the radial solutions and, on the way, develop the angular momentum formalism.