# Physical significance of othogonality condition?

what is the physical significance of othogonality condition? or what is meant when we say that two eigen states are orthogonal?

## Answers and Replies

George Jones
Staff Emeritus
Science Advisor
Gold Member
what is the physical significance of othogonality condition? or what is meant when we say that two eigen states are orthogonal?

Here's one thing. Suppose that two orthogonal eigstates of the observable A correspond to distinct eigenvalues a1 and a2. Perform a meausurement of A and assume that the result is a1. Perform a second measurement of A immediately after the first measurement. The probabilty that the result is a2 is zero.

Think about the postulates of quantum mechanics:

If I can measure certain eigenstates, the probability of measuring a particle in the state $$|\Psi\rangle$$ and obtaining it being in the state $$|\phi\rangle$$ after measurement is

$$P(\phi) = |\langle \phi | \Psi \rangle|^2$$

So what if these two states are orthogonal to each other?

reilly
Science Advisor
Orthogonal means perpendicular, uncorrelated. suppose A is orthogonal to B-- like spin up and spin down. Then they are independent in the sense that in a quantum measurement there's no A-B imterference.
Regards,
Reilly Atkinson

Except for energetic degeneracy, having an orthogonal set of solutions to the Schroedinger equation means that you've got wavefunctions corresponding to pure states. These pure states correspond to definite and different values of quantum numbers (say, orbital occupation) and have possibly distinct physical properties ( dipole moment, polarizability). Any experimental measurement of an eigenvalue property of a mixture of orthogonal states would measure an average of the quantity in the pure states. Also, having a pure state means that there is a non-zero probability to trap and observe the individual quantum states by themselves. For example, the lowest energy electronic configuration of helium is with the 1s orbital doubly occupied. And that's almost all of what's inside a helium gas cylinder. But there are orthogonal higher energy states, say 2s(2), that may have different values of, polarizability, for example. Contrary to what's above, you can observe different instantaneous values of a property of a pure state at different times. The proper value is given as a time average - the electron density is one example, because it's a probability distribution.
In the case of degeneracy, such as with unpaired electron spin not in a magnetic field, the situation is a little more complicated, because the degenerate orthogonal solutions can mix.
-Jim