Why Are Degenerate Quantum States Not Orthogonal?

[anilrapire]

would someone mind explaining why, in general, degenerate (quantum) states are not orthogonal?

 

[vanesch]

Degenerate states form a whole subspace. The linear superposition of two degenerate states is again a state with the same eigenvalue. So your question is equivalent to:
can someone explain me why all vectors aren't orthogonal ?

However, it is always possible to choose an orthogonal basis in this subspace, and those eigenvectors ARE orthogonal.

cheers,
Patrick.

 

[R0man]

Degenerate quantum states are not orthogonal because they have the same energy level or eigenvalue. In quantum mechanics, the energy of a system is represented by its eigenvalues, and the corresponding states are called eigenstates. These eigenstates are orthogonal to each other, meaning they are perpendicular and have no overlap in terms of their wavefunctions. This is an important property of quantum states as it allows for the accurate description and prediction of the behavior of a system.

However, when two or more states have the same energy level, they are considered degenerate states. This means that they cannot be distinguished based on their energy alone, and there may be multiple ways for the system to exist in that particular energy level. As a result, these degenerate states can have different wavefunctions, but still have the same energy.

Since degenerate states have different wavefunctions, they cannot be orthogonal to each other. This is because orthogonality requires the inner product of two states to be zero, but if the two states have different wavefunctions, their inner product will not be zero. This is why degenerate quantum states cannot be orthogonal.

Furthermore, degeneracy also violates the principle of superposition, which states that any quantum state can be represented as a linear combination of other states. However, in the case of degeneracy, there is no unique way to represent the state as a linear combination of other states, making it impossible to determine the probability of the system being in a particular state.

In summary, degenerate quantum states are not orthogonal because they have the same energy level and therefore, cannot be distinguished based on energy alone. This leads to different wavefunctions, violating the principle of orthogonality and superposition.