In quantum mechanics, two quantum states are said to be orthogonal if their wave functions are mutually perpendicular. More formally, given two wave functions ψ₁ and ψ₂, they are orthogonal if the inner product (or dot product) of the two wave functions is zero:
∫ ψ₁*(x) ψ₂(x) dx = 0
Here, ψ₁*(x) represents the complex conjugate of ψ₁(x), and the integral is taken over the relevant space.
To prove that two wave functions are orthogonal if their wave functions are mutually perpendicular, you can follow these steps:
Start with the assumption that the wave functions ψ₁ and ψ₂ are mutually perpendicular, meaning their inner product is zero.
Take the inner product of ψ₁*(x) and ψ₂(x) by integrating their product over the appropriate space. This integral will give you the result of the inner product.
If the integral evaluates to zero, then you have proven that the wave functions ψ₁ and ψ₂ are orthogonal.
The proof relies on the definition of orthogonality, which states that if the inner product of two wave functions is zero, then the states are orthogonal.
It's important to note that orthogonal states have several important properties in quantum mechanics. For example, in a system with orthogonal states, the probability of measuring one state is independent of the presence or absence of the other orthogonal states. Orthogonal states are commonly used as a basis for representing quantum states, and they play a crucial role in various quantum algorithms and calculations.