3.3 The Operators of Quantum Mechanics

The numerical quantities that the old Newtonian physics uses, (position,
momentum, energy, ...), are just shadows

of what
really describes nature: operators. The operators described in this
section are the key to quantum mechanics.

As the first example, while a mathematically precise value of the
position of a particle never exists, instead there is an -position operator . It turns the wave
function into :

(3.3) |

Instead of a linear momentum , there is an -momentum operator

(3.4) |

(3.5) |

The factor in makes it a Hermitian operator (a proof of that is in derivation {D.9}). All operators reflecting macroscopic physical quantities are Hermitian.

The operators and are defined similarly as :

(3.6) |

The kinetic energy operator is:

(3.7) |

This is an example of the “Newtonian analogy”: the relationships between the different operators in quantum mechanics are in general the same as those between the corresponding numerical values in Newtonian physics. But since the momentum operators are gradients, the actual kinetic energy operator is, from the momentum operators above:

(3.8) |

Mathematicians call the set of second order derivative operators in
the kinetic energy operator the Laplacian

, and
indicate it by :

(3.9) |

(3.10) |

Following the Newtonian analogy once more, the total energy operator, indicated by , is the the sum of the
kinetic energy operator above and the potential energy operator :

(3.11) |

This total energy operator is called the Hamiltonian and it is very important. Its eigenvalues are
indicated by (for energy), for example , ,
, ...with:

(3.12) |

It is seen later that in many cases a more elaborate numbering of the eigenvalues and eigenvectors of the Hamiltonian is desirable instead of using a single counter . For example, for the electron of the hydrogen atom, there is more than one eigenfunction for each different eigenvalue , and additional counters and are used to distinguish them. It is usually best to solve the eigenvalue problem first and decide on how to number the solutions afterwards.

(It is also important to remember that in the literature, the Hamiltonian
eigenvalue problem is commonly referred to as the
time-independent Schrödinger equation.

However, this
book prefers to reserve the term Schrödinger equation for the
unsteady evolution of the wave function.)

Key Points

- Physical quantities correspond to operators in quantum mechanics.

- Expressions for various important operators were given.

- Kinetic energy is in terms of the so-called Laplacian operator.

- The important total energy operator, (kinetic plus potential energy,) is called the Hamiltonian.