Principles of Quantum Mechanics
Before a detailed study of quantum mechanics, it is worth introducing some of the mathematical terms and concepts that will feature heavily in this area, and may otherwise be new or unfamiliar:
The most trivial point to note is that in quantum mechanics, the quantity h/2π is frequently encountered. A purely notational change is to designate this quantity as . This is done merely to simplify the appearance of some of the equations that we shall encounter.
The concept of an operator is of great importance in quantum mechanics.
An operator is an instruction to carry out a mathematical operation upon the following mathematical function.
Thus for example in this equation :
the operator is and the operation is to differentiate with respect to x.
This is an example of a differential operator, as the operation involves differentiation of the function. Another important class of operators is the multiplicative operators, such as x. This is an operator, as it is an instruction to multiply the following function by x.
We describe an equation of the following type as an eigenvalue equation:
or more simply, where Ω represents a general operator, ψ a general function and ω a constant:
A function that obeys such relation is said to be an eigenfunction of the operator Ω. The constant ω is called the eigenvalue of this eigenfunction.
The operator is d/dx. The function is e4x, and the answer is 4e4x. Thus this is an eigenvalue equation. The eigenfunction is e4x , and the eigenvalue is 4.
is not an eigenvalue equation, as the original function (ex2) is multiplied by another function of x (2x).
One particularly important operator in quantum mechanics is the Hamiltonian operator, H. The precise form of this operator varies according to the system under consideration, but in all cases it is the operator corresponding to the total energy of the system. i.e. if we can find a mathematical function that precisely describes the system of interest, then if it is an eigenfunction of the Hamiltonian operator, the eigenvalue of the function is equal to the total energy of the system:
where Ψ is an eigenfunction that describes the system, and E is the total energy of the system.
Note that the above equation is a general form of a very important equation known as the Schrodinger equation. This equation is of great importance in quantum mechanics, because it provides a way of calculating the function Ψ.
We must write out the equation as above, substitute in the full form of the Hamiltonian operator for the system under consideration, and then solve the equation to obtain Ψ. Examples of this procedure occur throughout quantum mechanics.
Since H is the operator corresponding to the total energy of the system, it should come as no surprise to learn that it is the sum of the operators corresponding to the kinetic and potential energies of the system.
It is also appropriate to introduce here the explicit forms of two operators that correspond to observables.
This phrase means that eigenvalues of the operator give the value of a measurable physical property of the system.
i.e. There is an operator corresponding to linear momentum. If application of this operator to a function that describes the system yields an eigenvalue, then this is the value of the linear momentum of the system. (If the function that describes the system is not an eigenfunction of the operator, then there is no definite value of the linear momentum; it cannot be precisely measured.)
The operator corresponding to position along the x axis is simply x, multiplication by x. (The position operators in the y and z directions are, analogously, multiplication by y or z respectively.)
The operator corresponding to linear momentum in the x direction is
i.e. differentiation with respect to x followed by multiplication by / i. (i is the square root of -1). Linear momenta in the y and z directions are obtained from analogous expressions in y and z.
Operators corresponding to other observables may be constructed from these two by substituting them for the position or momentum in classical expressions. e.g. the classical expression for kinetic energy is p2/2m. Thus to obtain the operator for kinetic energy, we write out that it is p2/2m. We can substitute in for p to give the explicit form of the kinetic energy operator in one dimension:
Thus the kinetic energy in one dimension is obtained by taking a function which describes the whole system, differentiating it twice with respect to x and multiplying the result by –2 / 2m.
Note that the second derivative of a function is a measure of the rate of change of its gradient, or loosely speaking the curvature of the function.
This implies a direct relationship between the curvature of a function and the kinetic energy of a system described by the function.
Thus the curvature of a graph of some function which describes a system can be used to make a rough estimate of the kinetic energy of the system.
The expectation value of an operator, denoted < W >, is defined as follows:
where Ψ is a normalised wavefunction of the system of interest. This is a function which completely describes the system (see the next page on wavefunctions for details).
The expectation value of an operator that corresponds to an observable is the mean value of that observable for a large number of observations upon the system. i.e. if Ω is the kinetic energy operator, then the mean value of a large number of measurements of the kinetic energy of the system will be equal to the expectation value of the operator.
Note that if the function Ψ is an eigenfunction of the operator, the expectation value reduces to the eigenvalue of the function.