vaihetekijä

A vaihetekijä, or phase factor, is a complex number of magnitude one used in various areas of physics and engineering, particularly in signal processing and quantum mechanics. It represents a rotation in the complex plane, effectively shifting the phase of a signal or a wavefunction without altering its amplitude. Mathematically, a vaihetekijä is typically expressed in the form e^{i\phi}, where 'i' is the imaginary unit and 'φ' is the phase angle in radians. This exponential form is a direct consequence of Euler's formula, which relates the exponential function to trigonometric functions: e^{i\phi} = \cos(\phi) + i\sin(\phi).

In signal processing, vaihetekijät are crucial for manipulating signals in the frequency domain. For example, they are used in Fourier transforms and digital filtering to adjust the timing or alignment of different frequency components. Multiplying a signal by a vaihetekijä corresponds to a phase shift in the time domain. In quantum mechanics, vaihetekijät play a fundamental role in describing the evolution of quantum states. The time evolution of a quantum system is governed by the Schrödinger equation, and its solutions often involve multiplying the state vector by a time-dependent vaihetekijä, which represents the accumulation of phase over time. The physical interpretation of these phase factors can be subtle, as only relative phase differences between quantum states are typically observable.