The Plancherel theorem in quantum mechanics is a mathematical tool used to describe the relationship between a function in real space and its corresponding function in Fourier space. It states that the integral of the squared modulus of a function in real space is equal to the integral of the squared modulus of its Fourier transform.
In quantum mechanics, the Plancherel theorem is used to describe the properties of a quantum system in terms of its wave function and its corresponding momentum-space wave function. This allows for the calculation of important quantities such as the energy spectrum and the probability distribution of a quantum system.
The proof of the Plancherel theorem for quantum systems involves using the properties of the Fourier transform and the wave function to show that the integral of the squared modulus of the wave function in real space is equal to the integral of the squared modulus of its Fourier transform. This proof relies on the assumption that the wave function is square-integrable.
The Plancherel theorem is significant in quantum mechanics as it allows for the translation of important physical quantities from real space to momentum space. This is particularly useful in the study of quantum systems, as it provides a way to analyze the properties of a system in terms of its wave function and its corresponding momentum-space wave function.
The Plancherel theorem has some limitations in its application to quantum systems. One limitation is that it only applies to systems with a finite number of dimensions, making it less useful for studying systems with an infinite number of dimensions. Additionally, the theorem assumes that the wave function is square-integrable, which may not be the case for all quantum systems.