Yes, the wave-like manifestation of an electron, characterized by its wavelength, frequency, and other wave properties, is distinct from the mathematical wavefunction used to describe the probability distribution of the electron's observables.
The wave-like behavior of an electron refers to its ability to exhibit phenomena such as interference and diffraction, similar to classical waves. This behavior is described by its de Broglie wavelength, which is related to the momentum of the electron. The wavelength and frequency of an electron are inversely proportional, following the relationship λ = h/p, where λ is the wavelength, h is Planck's constant, and p is the momentum.
On the other hand, the wavefunction of an electron is a mathematical function that encodes the probabilistic nature of quantum mechanics. It provides information about the possible states that an electron can occupy and the probabilities associated with each state. The wavefunction is a complex-valued function that depends on the position and time, and its squared magnitude gives the probability density of finding the electron in a particular state.
The wavefunction can be used to calculate expectation values of observables, such as position, momentum, and energy, through mathematical operations called operators. These operators act on the wavefunction to extract information about the observables of interest. The wavefunction itself does not directly represent the physical properties of the electron but rather the probabilities associated with those properties.
In summary, the wave-like manifestation of an electron refers to its wave properties, such as wavelength and frequency, which are related to its momentum. The wavefunction, on the other hand, is a mathematical representation of the probability distribution of the electron's observables, such as position and momentum, and it is not directly related to the wave properties of the electron.