III. Exactly Solvable Problems The systems for which exact QM solutions can be found are few in number and they are not particularly interesting in and of themselves; typical experimental systems are much more complicated than any exactly solvable Hamiltonian. However, we must understand these simple problems if we are to have any hope of attacking more complicated systems. For example, we will see later that one can develop very accurate approximate solutions by examining the difference between a given Hamiltonian and some exactly solvable Hamiltionian. The two problems we will deal with here are: the Harmonic Oscillator and Piecewise Constant Potentials. A. Operators and States in Real Space To solve even simple one-dimensional problems, we need to be able to describe experiments in real space in terms of Hilbert space operators. The ansatz is fairly straightforward. A generic (classical) observable can be associated with some function of the position and momentum: ),(qpA Note that we assume (for now) that there is only one particle, so the observable only depends on one position and one momentum. The associated quantum operator is obtained by replacing the classical variables ),(qp with the corresponding quantum operators )ˆ,ˆ( qp : )ˆ,ˆ(ˆqpAA = The position and momentum operators satisfy the canonical commutation relation: []ipq=ˆ,ˆ This association has a deep connection with the role of Poisson brackets in classical mechanics. Unfortunately, this connection is completely lost on the typical chemist, who is unfamiliar with Poisson brackets to begin with. Now, the fact that pˆ and qˆ do not commute poses an immediate problem. What if we want to associate a quantum operator with a classical observable likepqqpA=),(? We have more than one choice: we could choose qpAˆˆˆ= or pqAˆˆˆ= . For this simple productform, the dilemma is easily resolved by requiring Aˆ to be Hermitian, in which case the only possible choice is the symmetric form: ()pqqpAˆˆˆˆˆ21+= You can verify for yourself that this operator is, indeed Hermitian. When ),(qpA contains more complicated products of pˆ and qˆ (e.g. ()qpqpA −= 1cos),(3) the solution is, well, more complicated. In fact, there is no a general way to associate arbitrary (non-linear) products of pˆ and qˆ with a unique quantum operator. Fortunately, we will not be interested in classical observables that involve non-linear products of pˆ and qˆ in this course. Thus this is not a practical obstacle for us; however, there is an on-going debate within the physics community about how these products should be treated. B. The Harmonic Oscillator Classically, a Harmonic oscillator is a system with a linear restoring force, ()kqqV−=∇− It is easily verified that the correct potential in this case is: ()2221221qmkqqVω== mk=ω where k is the effective force constant and ω is the frequency of the oscillation. In QM, it is often more convenient to work in terms of this frequency, and this is what we will do. To begin with, we will be interested in the observable energies associated with this potential. In classical mechanics the total energy is generated by the Hamiltonian, which we can immediately associate with a quantum operator: 2221222212ˆ2ˆˆ2),( qmmpHqmmpqpHωω+=+= There are two main motivations for studying the harmonic oscillator. The first is that it has a deep relationship to many other exactly solvable problems in QM. You will sometimes even hear it said that all exactly solvable problems are the harmonic oscillator in disguise. This is because virtually every exactly solvable QM potential has some observable that evolves under a linear force; in the case of the harmonic oscillator, it is the position, but one can arrive at arbitrarilycomplicated (but still exactly solvable) potentials by using different linear evolutions. Another justification for studying harmonic potentials is that even for a very anharmonic potential, the potential looks harmonic if one is near enough the minimum. Thus, if we expand around the minimum (and choose our zero of energy so that ()00≡V ): ( )2022212022210... qqVqqVqqVqVδδδδ∂∂≈+∂∂+∂∂= And we see that the potential is approximately harmonic. At this point, it is convenient to convert to reduced units by choosing our units of length, mass and energy so that 1===ωm. These units are merely out of convenience and in the end, once we have calculated an observable (such as the position) we will need to convert the result to a set of standard units (such as meters). We can do this by noting that, in reduced units: ωmLength=1: ω=1:Energy The main benefit at the moment is that it removes the relatively unimportant factors of , m and ω from our equations, so that in natural units: 2ˆ2ˆˆ22qpH += Notice that this is the most we can do; there are only three fundamental units, and so if there were a fourth constant (an anharmonicity, say) we would not be able to scale away this unit. Later, we will also very often be interested in only fixing some of our units (for example 1==m) and leaving others free. This will greatly simplify the algebra in many instances. Now, define two operators: ()piqaˆˆˆ21+= and ()piqaˆˆˆ21†−= As the notation suggests, these operators are Hermitian conjugates of one another. However, they do not commute []()()[][][][][]()[ ] [ ]1ˆ,ˆˆ,ˆˆ,ˆˆ,ˆˆ,ˆˆ,ˆˆˆ,ˆˆˆ,ˆ21212121†=−=+−+=−+=pqiqpipppqiqpiqqpiqpiqaa Now, we can re-write the Hamiltonian in terms of these operators if we notice that 0 0()†21ˆˆˆaaq += and ()†2ˆˆˆaapi−=− Then, ()()( ) ( )( )††21††††††††2†2†22ˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆ4ˆˆ4ˆˆ2ˆ2ˆˆaaaaaaaaaaaaaaaaaaaaaaaaqpH+=+++++−−−=++−−=+= We now make use of the commutation relations to write aaaaaaaaˆˆ1ˆˆ1ˆˆˆˆ††††+==− which leads to ()()21†††21††21ˆˆˆˆ1ˆˆˆˆˆˆˆ+=++=+= aaaaaaaaaaH This leads us to our first key point: the eigenvalues of the harmonic oscillator Hamiltonian are all positive. To see this, note that for any state, ψ, the average energy will be: 21221†ˆˆˆˆ+=+=ψψψψψψψaaaH Since the norm of a vector is always positive, we conclude the average energy is always positive. However, for an eigenstate the average energy is just the energy eigenvalue and the point is made. Next, define the number