Ladder operators

Ladder operators

In linear algebra (and its application to quantum mechanics), a raising or lowering operator (collectively known as ladder operators) is an operator that increases or decreases the eigenvalue of another operator. In quantum mechanics, the raising operator is sometimes called the creation operator, and the lowering operator the annihilation operator. Well-known applications of ladder operators in quantum mechanics are in the formalisms of the quantum harmonic oscillator and angular momentum.

Suppose that two operators X and N have the commutation relation: [N,X] = cX,for some scalar "c". Then the operator "X" will act in such a way as to shift the eigenvalue of an eigenstate of "N" by "c":

In other words, if |n angle is an eigenstate of N with eigenvalue "n" then X|n angle is an eigenstate of "N" with eigenvalue "n" + "c". A "raising operator" for "N" is an operator "X" for which "c" is real and positive and a "lowering operator" is one for which "c" is real and negative.

If "N" is a Hermitian operator then "c" must be real and the Hermitian adjoint of "X" obeys the commutation relation:: [N,X^dagger] = -cX^dagger.,In particular, if "X" is a lowering operator for "N" then "X"† is a raising operator for "N" (and vice-versa).


Wikimedia Foundation. 2010.

Игры ⚽ Нужно решить контрольную?

Look at other dictionaries:

  • Ladder logic — is a philosophy of drawing electrical logic schematics. It is now a graphical language very popular for programming Programmable Logic Controllers (PLCs). It was originally invented to describe logic made from relays. The name is based on the… …   Wikipedia

  • Creation and annihilation operators — Quantum optics operators Ladder operators Creation and annihilation operators Displacement operator Rotation operator (quantum mechanics) Squeeze operator Anti symmetric operator Quantum corr …   Wikipedia

  • Quantum harmonic oscillator — The quantum harmonic oscillator is the quantum mechanical analogue of the classical harmonic oscillator. It is one of the most important model systems in quantum mechanics because an arbitrary potential can be approximated as a harmonic potential …   Wikipedia

  • Anti-symmetric operator — In quantum mechanics, a raising or lowering operator (collectively known as ladder operators) is an operator that increases or decreases the eigenvalue of another operator. In quantum mechanics, the raising operator is sometimes called the… …   Wikipedia

  • Phonon — For KDE Software Compilation 4 s multimedia framework, see Phonon (KDE). Normal modes of vibration progression through a crystal. The amplitude of the motion has been exaggerated for ease of viewing; in an actual crystal, it is typically much… …   Wikipedia

  • Quantum field theory — In quantum field theory (QFT) the forces between particles are mediated by other particles. For instance, the electromagnetic force between two electrons is caused by an exchange of photons. But quantum field theory applies to all fundamental… …   Wikipedia

  • Displacement operator — Quantum optics operators Ladder operators Creation and annihilation operators Displacement operator Rotation operator (quantum mechanics) Squeeze operator Anti symmetric operator Quantum corre …   Wikipedia

  • Laplace–Runge–Lenz vector — Throughout this article, vectors and their magnitudes are indicated by boldface and italic type, respectively; for example, left| mathbf{A} ight| = A. In classical mechanics, the Laplace–Runge–Lenz vector (or simply the LRL vector) is a vector… …   Wikipedia

  • Squeeze operator — The squeeze operator for a single mode is:hat{S}(z) = exp left ( {1 over 2} (z^* hat{a}^2 z hat{a}^{dagger 2}) ight ) , qquad z = r e^{i heta}where the operators inside the exponential are the ladder operators. The squeeze operator is ubiquitous… …   Wikipedia

  • Coherent states in mathematical physics — Coherent states have been introduced in a physical context, first as quasi classical states in quantum mechanics, then as the backbone of quantum optics and they are described in that spirit in the article Coherent states (see also [1]). However …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”