Application To Quantum Mechanics
In quantum mechanics, rotational invariance is the property that after a rotation the new system still obeys Schrödinger's equation. That is
- = 0 for any rotation R.
Since the rotation does not depend explicitly on time, it commutes with the energy operator. Thus for rotational invariance we must have = 0.
Since = 0, and because for infinitesimal rotations (in the xy-plane for this example; it may be done likewise for any plane) by an angle dθ the rotation operator is
- R = 1 + Jz dθ,
- = 0;
thus
- d/dt(Jz) = 0,
in other words angular momentum is conserved.
Read more about this topic: Rotational Invariance
Famous quotes containing the words application to, application, quantum and/or mechanics:
“Preaching is the expression of the moral sentiment in application to the duties of life.”
—Ralph Waldo Emerson (1803–1882)
“It is known that Whistler when asked how long it took him to paint one of his “nocturnes” answered: “All of my life.” With the same rigor he could have said that all of the centuries that preceded the moment when he painted were necessary. From that correct application of the law of causality it follows that the slightest event presupposes the inconceivable universe and, conversely, that the universe needs even the slightest of events.”
—Jorge Luis Borges (1899–1986)
“But how is one to make a scientist understand that there is something unalterably deranged about differential calculus, quantum theory, or the obscene and so inanely liturgical ordeals of the precession of the equinoxes.”
—Antonin Artaud (1896–1948)
“It is only the impossible that is possible for God. He has given over the possible to the mechanics of matter and the autonomy of his creatures.”
—Simone Weil (1909–1943)