By James Lawniczak and Mike Payne
The Basics
Basic Terminology:
Operator: A set of instructions on how to change a function.^{[3]}
Eigenfunction and Eigenvalue: You have an operator F, which acts on a function g. If you get
F(g) = k(g), where k is just a constant times our function. The constant k is your eigenvalue and g is an eigenfunction.
Wavefunction: First, a wavefunction Ψ(x) is a function dependent on position and it encapsulates all that we know about a particle. It can take on both positive and negative values. Squaring the wavefunction gives
Ψ(x)^{2}, which only has positive values. The area under the curve of Ψ(x)^{2} between two values of x gives the probability of finding a particle between those two x values. ^{[1]}.
What is the Hamiltonian and where does it come from?
The quantum Hamiltonian is an operator that can be used to find the allowed energy values of a physical system. It contains everything we know about the physical system. In addition, the Hamiltonian is a key component in the Schrodinger equation, which tells us how the physical system evolves with time. The quantum Hamiltonian is taken from the classical Hamiltonian equation H(p,x) = (p^2)/2m+V(x) ^{[5]}. In short, the momentum and potential energy function are converted to an operator, and so we have the quantum Hamiltonian:
where H,P, and V are the Hamiltonian, momentum operator, and potential energy operator, respectively^{[1]}.
How is the quantum Hamiltonian operator different from the classical Hamiltonian function?
Remember that the Hamiltonian tells us how to find out the energy of the physical system you are working with (atom, particleinabox, etc.). The way classical physics approaches this is to define the Hamiltonian to be a function that depends on momentum, position, and any other important variables, and the resulting value of that function is the energy of the system^{[5]}. Quantum physics defines the Hamiltonian as an operator whose input is a wavefunction. The eigenvalues of the Hamiltonian are the allowed energy values. So the classical Hamiltonian is just a function of variables like position and momentum, while the quantum Hamiltonian has eigenvalues that tell us the possible energy levels of the physical system. ^{[3]}:
What is the Schrodinger Equation, and how is it related to the Hamiltonian?
The Schrodinger equation is a way of modeling quantum behavior by a wave function, represented in the Schrodinger equation by Ψ. There are two different forms of the Schrodinger equation  a timeindependent and a timedependent form.
The timeindependent Schrodinger equation is given as follows:
This equation gives the relationship between the 2nd derivative of Ψ(x) and the energy of the system E. However, the left side of the equation includes some familiar terms that turn out to be the kinetic energy operator from above  because of this, the equation can be rewritten in operator form as:
Since the Hamiltonian is the sum of the kinetic and potential energy operators, as discussed above, the Hamiltonian operator can thus be used in this system to relate a wavefunction and its total energy:
Now, when given a wavefunction, it is possible to use the Hamiltonian operator to calculate the possible energeies in a given quantum system. It can be seen that a wavefunction which corresponds to a solution of the Schrodinger Equation is an eigenfunction of the Hamiltonian operator, meaning that when the Hamiltonian operator is applied to the Schrodinger equation, the solution is some constant (termed an eigenvalue) multiplied by the original wavefunction. In this context, the eigenvalue correspond to allowed energy states of the quantum system.^{[6]}
Ideas to Ponder
Remember that the quantum Hamiltonian is expressed as:
where H,P,V are quantum operators.
Notice that potential energy and kinetic energy term can be defined in different ways depending on what forces are present in your system. For example, the potential energy term can be dependent on whether there is an electric field. Thus, the specific Hamiltonian you will use in your calculations will depend on the the physical system you are working with.
Some questions to consider:
How is the Hamiltonian expressed in various physical systems, such as particle in a box, the rigid rotor model, and the simple harmonic oscillator?
What is the Schrodinger equation in each of these situations?
iClicker Questions:
True or False:
The quantum Hamiltonian gives us information about the exact energy of a physical system given its wavefunction.
(a) True, since we can plug in the different parameters of the state (like position, time, etc.) and get an exact value from the Hamiltonian.
(b) True, since we can get the eigenvalues of the Hamiltonian and so calculate the exact energy of a system.
(c) False, since the quantum Hamiltonian gives us only the allowed energies for a system, and not necessarily the exact energy for the system.
(d) False, since the quantum Hamiltonian is used for the Schrodinger equation, and not for calculating energies of a physical system.
Answer: C
The differential operator D(f) takes the derivative of the function f. Thus, the function f = x^{2} is an eigenfunction of D and has eigenvalue 2.

True, since the derivative of x^{2}is 2x, and so the eigenvalue is 2.

True, since the derivative of x^{2 }is 2x, which is an eigenfunction of D.

False, since 2 is not an eigenvalue of D.

False, since the derivative of x^{2 }is 2x, which is not an eigenfunction.
Answer: D
The Hamiltonian is in the same form in both the timedependent and timeindependent Schrodinger equation (the timeindependent Schrodinger equation tells us how a quantum system evolves in time).

True, since the Hamiltonian is the operator that has all the information about the physical system.

True, since the total energy of a system doesn’t depend on time, the separation into the time evolution and the timeindependent Schrodinger equation won’t affect the function of the Hamiltonian.

False, when the Schrodinger equation is no longer dependent on time, the potential energy is able to change therefore changing the Hamiltonian.

False, since the Hamiltonian is a specific value, the presence of time affects its outcome.
Answer: A
If the Hamiltonian and the wavefunction is the same for two systems, then the systems will be measured to have the same energy values measured.
a) True, since the Hamiltonian is identical, the wavefunctions will evolve identically according to the Schrodinger equation, resulting in identical measurements for energy.
b) True, since plugging in time for the Hamiltonian will give the same values for energy at any time.
c) False, since measuring the system may give different results for the energy due to the probabilistic nature of the wavefunction.
d) False, since the wavefunctions of the two systems evolve in a probabilistic manner.
Answer: C
Difficult Concept: Classical functions and quantum operators
Classical functions and quantum operators work in distinct ways, which changes the way in which they are used mathematically. Classical functions, like the classical Hamiltonian, give a value for the energy of the system in a state specified by its position, time, and momentum. For example, given some information (parameters like position, time, etc.) about a physical system, you just plug in these parameters in the classical Hamiltonian (which is a typical classical function), and the energy of the system will be known. However, for quantum operators, eigenvalues and eigenfunctions of an operator are what we focus on.
An eigenfunction of a certain operator is a function that, when used as an input for this operator, gives a nonzero number times the function as the output. The nonzero number is the eigenvalue associated with that eigenfunction that was plugged in. For example, the quantum Hamiltonian is an operator which tells us the possible energies of a physical system. The eigenvalues of the quantum Hamiltonian are the possible energies of a system.^{2,3}
True or False: Defining the Hamiltonian as a quantum operator is useful for modeling the discreteness of energy levels.
(a) True, since finding eigenvalues of an operator can give us discrete values.
(b) True, the eigenvalues of the Hamiltonian are the probabilities of the state of a system.
(c) False, because the Schrodinger equation needs quantum operators.
(d) False, the Hamiltonian only calculates specific values for the wave function not probabilities.
Answer: A
Hard Concept Within A Concept: The reason why the distinction between quantum operators and classical functions is confusing.
What made the distinction between classical functions and quantum operators confusing is that we are used to plugging in values to functions and operators (which act in a way like functions, except they are more general). Instead of considering the inputoutput properties of an operator, we need to consider other properties of the operator, namely, the operator’s eigenfunctions and eigenvalues.
True or False:
Quantum operators work in the same fashion as classical functions.
(a) True, since plugging in a value for a function always yields a number that is a constant multiple of the number you inputed.
(b) True, since you can get eigenfunctions for a classical function.
(c) False, because eigenfunctions are only applied to quantum physics.
(d) False, because there are no eigenvalues for a classical function.
Answer: D
Citations
[1] Griffiths, D. J. Introduction to Quantum Mechanics, 2nd ed.; Pearson Education: Upper Saddle River, 2005; , pp 23, 2627.
[2] HyperPhysics. The Operator Postulate. http://hyperphysics.phyastr.gsu.edu/hbase/quantum/qm2.html#c1(accessed Sept 15, 2014).
[3] Baggot, Jim The Meaning of Quantum Theory, 1992, Oxford university Press Inc., New York; pp 39, 4445.
[4] McQuarrie, Donald A., and John D. Simon. Physical Chemistry: A Molecular Approach. Sausalito, CA: U Science, 1997. Print.
[5] New York University. The Hamiltonian formulation of classical mechanics.
http://www.nyu.edu/classes/tuckerman/stat.mech/lectures/lecture_1/node4.html (accessed Sept. 29, 2014).
[6] Oxtoby, D. W.; Gillis, H. P.; Campion, A. Principles of Modern Chemistry, Seventh Edition; Cengage Learning: Belmont, 2012; pp 169171.
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