# Physical significance of othogonality condition?

• sruthisupriya
In summary, the physical significance of the orthogonality condition is that it allows for pure states to exist and be observed separately. This condition also allows for the measurement of average values of different properties of these pure states. In the case of degeneracy, the orthogonal solutions may mix, making the measurement of individual states more complicated.

#### sruthisupriya

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

sruthisupriya said:
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?

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

## 1. What is the physical significance of orthogonality in science?

The concept of orthogonality is important in many areas of science, including physics, engineering, and mathematics. It refers to the relationship between two vectors or functions that are perpendicular to each other, meaning they form a 90-degree angle. This relationship has physical significance because it allows us to simplify complex systems and analyze them in a more efficient and precise manner.

## 2. How does orthogonality relate to symmetry and conservation laws?

In physics, symmetries are closely related to orthogonality. This is because symmetries can be described mathematically as transformations that preserve the orthogonality of a system. Conservation laws, such as the law of conservation of energy, also rely on orthogonality. For example, in a closed system, the total energy remains constant because the energy is perpendicular to any possible changes in the system.

## 3. Can you give an example of how orthogonality is used in physics?

One example of how orthogonality is used in physics is in quantum mechanics. In this field, the wave function of a particle is described as a vector in a multi-dimensional space. The orthogonality of these wave functions allows us to determine the probability of finding a particle in a certain location. The wave functions of different particles are also orthogonal, meaning they do not overlap, allowing us to distinguish between different particles in a system.

## 4. What is the significance of the orthogonality condition in Fourier analysis?

In Fourier analysis, the orthogonality condition is used to decompose a function into its component frequencies. This allows us to represent complex signals as a combination of simpler, orthogonal signals. This has numerous applications in science, such as signal processing, image compression, and solving differential equations.

## 5. How does orthogonality contribute to the accuracy and efficiency of scientific models?

Orthogonality plays a crucial role in the accuracy and efficiency of scientific models. By simplifying complex systems and reducing the number of variables, we can more accurately model and predict the behavior of the system. This leads to more efficient and precise calculations, allowing us to make better-informed decisions and advancements in science and technology.