The Hamiltonian 

 

The Hamiltonian is an operator that models the total energy in the system. Using the Hamiltonian operator, we can operate on the wavefunction describing the state of the system and solve for the energy of the system as a constant eigenvalue. The Hamiltonian is a sum of the potential energy operator and the kinetic energy operator. 

 

What is an operator?

 

                An operator applies a function on a system to create a new function. It is basically a way to change the values in a system. You have been familiar with operators for quite some time. Basic addition and subtraction, as well as multiplication and division, are operators. Imagine you have a system ψ. Any change on this system ψ would be considered an operation – even ψ + 1 is an operation on the function ψ. The Hamiltonian is just a more complex modification of a function.

                Quantum mechanics is built upon the concept of operators. Essentially, the wavefunction ψ does not model a specific attribute of a quantum function, such as position or momentum, but is an abstract representation of the quantum particle or wave, whichever you prefer to think of it as, itself. We can operate on the function as follows:

 

 

Now, we have the wavefunction operated on by the position operator. We cannot find the position of the particle discretely, but the average position instead. That we solve for through the expectation value:

 

 

The operator is dependent on the model of quantum mechanics that the function is being performed in, meaning that as the models of a particle change, the Hamiltonian operator on a wavefunction also changes, resulting in different expressions for energy. The basic time-independent Schrödinger equation describes the energy of a particle in a steady state as an eigenvalue with the Hamiltonian as the operator and the wavefunction as the eigenfunction:

 

 

where ψ is the actual wavefunction, which defines the particle, h is Planck's constant, and m is the mass of the particle. As defined above, the Hamiltonian is the total energy of the system. The total energy term is the kinetic energy of the system as defined by quantum mechanics, while V(x) represents the potential energy of the system. Based on the Schrödinger equation, we can factor out the term ψ(x), and then define the Hamiltonian as follows.

 

Thus, we have the definition of the Hamiltonian in a stationary state. It is the kinetic energy of a particle plus the potential energy of the system times the wavefunction. The Hamiltonian, however, is a special type of operator involving eigenvalues and eigenfunctions.

 

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Eigenvalues and Eigenfunctions

                 

 

Matrices have special properties known as eigenvalues and eigenfunctions. It is possible to write an operator as a matrix and a wavefunction as a matrix vector, which results in being able to use linear algebra, the math of matrices, to determine eigenvalues and eigenfunctions. Here we present a simplified form of eigenvalues to help understand the background of this function. Imagine a matrix A with r rows and c columns. Multiplying this matrix with an eigenvector x (a matrix with 1 column and r rows) gives a scalar λ multiplied by the eigenvector.

 

 

Subtracting λx from both sides, and performing algebra yields:

 

 

where I is the identity matrix with rows r and columns c. The Hamiltonian is part of a subclass of eigenfunctions known as Hermitian eigenfunctions because:

 

 

 From the knowledge that this function is a Hermitian eigenfunction, we can make some generalizations.

1. All eigenvalues are real. This is important because it guarantees real values for all energy values.
2. The eigenfunctions of different eigenvalues are orthogonal.
3. The complete set of eigenfunctions can always be chosen as orthonormal. 

 

Most of the implications of these statements are beyond the scope of this summary, but they are important in describing the necessity that the Hamiltonian is a Hermitian eigenfunction.

 

     One common misconception about the Hamiltonian is that it only applies to systems within quantum mechanics. This is not true. The Hamiltonian is absolutely applicable to a classical mechanics system. Simply put, the Hamiltonian is the total energy of a system, which is defined in classical mechanics as the sum of the potential energy and the kinetic energy of the system being observed in terms of position and momentum:

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The Particle in a Box Model

     We can apply the idea of the Hamiltonian to various models of quantum mechanics. Imagine you have a box of length L. In order to simplify the system, we define the potential energy outside the box must be infinite, and inside the box as 0. The particle cannot move beyond the boundary of the box. This models distinctions between classical and quantum mechanics. Outside of the well ψ(x) is 0, meaning the particle can never be outside the box. Since the potential energy is 0 and the kinetic energy does not change, the Schrödinger equation becomes:

 

 

giving the Hamiltonian. There is no change except for the removal of the potential energy term for the particle when it is inside the box. For all possible positions:

 

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The Simple Harmonic Oscillator Model

     This model is essentially modeling the wave function according to Hooke's Law, F = -kx, where k is the spring constant. Based on Hooke's law and the fact that force is the negative derivative of potential energy, the potential energy becomes:

 

 

And the kinetic energy term stays the same. Thus the Schrödinger equation becomes:

 

giving the Hamiltonian. Nothing changes except for the potential energy term.

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The Rigid Rotor Model

     This model is essentially modeling the wave function as a rotating model, which changes both the potential and the kinetic energies. In the standard model we have been studying, we have been using the kinetic energy related to linear momentum p:

 

 

But, in the rigid rotor model, we use the angular momentum L for p, resulting in :

 

 

where μ is the apparent mass. The potential energy is defined as 0 because there is no force acting on the system, which causes no potential energy

 

Thus, the Schrödinger equation is:

 

 

Both the kinetic and potential energies changed.

 

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Problem Set (Answers after the Citations)

1. Calculate the classical Hamiltonian of an electron moving at 2.87 x 10⁴ m/s, toward a proton 1.5 nm away.

 

     2. Given the rigid rotor model of quantum mechanics, how does the kinetic energy change? The potential energy? The Hamiltonian?

 

     3. Explain why the fact that there is no potential energy in the rigid rotor is a consequence having no forces on the system.

 

Citations:

 

Bretscher, Otto. Linear Algebra with Applications. 3E.

Geva, Eitan. Chapter 2 Notes. CHEM 461.

Geva, Eitan. Chapter 3 Notes. CHEM 461.

Geva, Eitan. F12-Chem-260-L06 quantum laws. CHEM 260.

Griffiths, David, J. Introduction to Quantum Mechanics. 2E. Sections 2.2, 2.3, 2.4.

Oxtoby, David W.; Gills, H. Pat; Campion, Alan. Principles of Modern Chemistry. 7E. Section 4.5.

Sension, Roseanne. F12-Chem-260-L06 quantum laws-AM. CHEM 260.

 

Answers:

 

1. 

2. The kinetic energy becomes (L)²/(2μ), the potential energy 0, and the Hamiltonian equivalent to the kinetic energy.