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- Thread starter sruthisupriya
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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.

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sruthisupriya said:

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.

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

[tex] P(\phi) = |\langle \phi | \Psi \rangle|^2 [/tex]

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

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Reilly Atkinson

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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

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.

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.

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.

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.

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.

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