Properties
The displacement operator is a unitary operator, and therefore obeys, where I is the identity matrix. Since, the hermitian conjugate of the displacement operator can also be interpreted as a displacement of opposite magnitude . The effect of applying this operator in a similarity transformation of the ladder operators results in their displacement.
The product of two displacement operators is another displacement operator, apart from a phase factor, has the total displacement as the sum of the two individual displacements. This can be seen by utilizing the Baker-Campbell-Hausdorff formula.
which shows us that:
When acting on an eigenket, the phase factor appears in each term of the resulting state, which makes it physically irrelevant.
Read more about this topic: Displacement Operator
Famous quotes containing the word properties:
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)