In quantum mechanics, the inner product of two states corresponds to the probability amplitude due to the mathematical framework of quantum mechanics known as wave function or state vector formalism. This formalism describes the behavior of quantum systems using wave functions, which are complex-valued functions that contain information about the system's properties.
The inner product, also called the scalar product or dot product, is a mathematical operation between two vectors that produces a scalar quantity. In quantum mechanics, the inner product is used to calculate the probability amplitude when considering the superposition of quantum states.
In the wave function formalism, a quantum state is represented by a wave function, often denoted as Ψ (psi). The wave function contains all the information about the quantum system, including its position, momentum, and other observable quantities. The probability of observing a particular outcome, such as a particle being in a specific state or having a certain measurement result, is related to the square of the absolute value of the probability amplitude.
The probability amplitude is obtained by taking the inner product of two wave functions representing different quantum states. Mathematically, if Ψ₁ and Ψ₂ are the wave functions of two states, their inner product is denoted as ⟨Ψ₁|Ψ₂⟩. The probability amplitude, often represented as A, is obtained by taking the inner product and normalizing it:
A = ⟨Ψ₁|Ψ₂⟩ / (‖Ψ₁‖ * ‖Ψ₂‖)
Here, ‖Ψ₁‖ and ‖Ψ₂‖ are the norms of the respective wave functions. The square of the absolute value of the probability amplitude |A|² represents the probability of transitioning from one state to another, or the probability of observing a specific outcome.
The interpretation of the probability amplitude stems from the Born rule, which states that the probability of obtaining a particular outcome in a measurement is proportional to the square of the absolute value of the probability amplitude. This rule has been empirically confirmed through numerous experiments in quantum mechanics.
Therefore, the inner product of two states in quantum mechanics provides the mathematical framework for calculating probability amplitudes and determining the probabilities associated with different measurement outcomes.