Phase-space formulation | Wikipedia audio article
This is an audio version of the Wikipedia Article: https://en.wikipedia.org/wiki/Phase-space_formulation 00:01:37 1 Phase-space distribution 00:06:13 2 Star product 00:14:32 3 Time evolution 00:17:34 4 Examples 00:17:43 4.1 Simple harmonic oscillator 00:24:22 4.2 Free particle angular momentum 00:25:53 4.3 Morse potential 00:26:09 4.4 Quantum tunneling 00:30:26 4.5 Quartic potential 00:30:41 4.6 Schrödinger cat state 00:31:02 5 References Listening is a more natural way of learning, when compared to reading. Written language only began at around 3200 BC, but spoken language has existed long ago. Learning by listening is a great way to: - increases imagination and understanding - improves your listening skills - improves your own spoken accent - learn while on the move - reduce eye strain Now learn the vast amount of general knowledge available on Wikipedia through audio (audio article). You could even learn subconsciously by playing the audio while you are sleeping! If you are planning to listen a lot, you could try using a bone conduction headphone, or a standard speaker instead of an earphone. Listen on Google Assistant through Extra Audio: https://assistant.google.com/services/invoke/uid/0000001a130b3f91 Other Wikipedia audio articles at: https://www.youtube.com/results?search_query=wikipedia+tts Upload your own Wikipedia articles through: https://github.com/nodef/wikipedia-tts "There is only one good, knowledge, and one evil, ignorance." - Socrates SUMMARY ======= The phase-space formulation of quantum mechanics places the position and momentum variables on equal footing, in phase space. In contrast, the Schrödinger picture uses the position or momentum representations (see also position and momentum space). The two key features of the phase-space formulation are that the quantum state is described by a quasiprobability distribution (instead of a wave function, state vector, or density matrix) and operator multiplication is replaced by a star product. The theory was fully developed by Hilbrand Groenewold in 1946 in his PhD thesis, and independently by Joe Moyal, each building on earlier ideas by Hermann Weyl and Eugene Wigner.The chief advantage of the phase-space formulation is that it makes quantum mechanics appear as similar to Hamiltonian mechanics as possible by avoiding the operator formalism, thereby "'freeing' the quantization of the 'burden' of the Hilbert space". This formulation is statistical in nature and offers logical connections between quantum mechanics and classical statistical mechanics, enabling a natural comparison between the two (see classical limit). Quantum mechanics in phase space is often favored in certain quantum optics applications (see optical phase space), or in the study of decoherence and a range of specialized technical problems, though otherwise the formalism is less commonly employed in practical situations.The conceptual ideas underlying the development of quantum mechanics in phase space have branched into mathematical offshoots such as algebraic deformation theory (see Kontsevich quantization formula) and noncommutative geometry.
This is an audio version of the Wikipedia Article: https://en.wikipedia.org/wiki/Phase-space_formulation 00:01:37 1 Phase-space distribution 00:06:13 2 Star product 00:14:32 3 Time evolution 00:17:34 4 Examples 00:17:43 4.1 Simple harmonic oscillator 00:24:22 4.2 Free particle angular momentum 00:25:53 4.3 Morse potential 00:26:09 4.4 Quantum tunneling 00:30:26 4.5 Quartic potential 00:30:41 4.6 Schrödinger cat state 00:31:02 5 References Listening is a more natural way of learning, when compared to reading. Written language only began at around 3200 BC, but spoken language has existed long ago. Learning by listening is a great way to: - increases imagination and understanding - improves your listening skills - improves your own spoken accent - learn while on the move - reduce eye strain Now learn the vast amount of general knowledge available on Wikipedia through audio (audio article). You could even learn subconsciously by playing the audio while you are sleeping! If you are planning to listen a lot, you could try using a bone conduction headphone, or a standard speaker instead of an earphone. Listen on Google Assistant through Extra Audio: https://assistant.google.com/services/invoke/uid/0000001a130b3f91 Other Wikipedia audio articles at: https://www.youtube.com/results?search_query=wikipedia+tts Upload your own Wikipedia articles through: https://github.com/nodef/wikipedia-tts "There is only one good, knowledge, and one evil, ignorance." - Socrates SUMMARY ======= The phase-space formulation of quantum mechanics places the position and momentum variables on equal footing, in phase space. In contrast, the Schrödinger picture uses the position or momentum representations (see also position and momentum space). The two key features of the phase-space formulation are that the quantum state is described by a quasiprobability distribution (instead of a wave function, state vector, or density matrix) and operator multiplication is replaced by a star product. The theory was fully developed by Hilbrand Groenewold in 1946 in his PhD thesis, and independently by Joe Moyal, each building on earlier ideas by Hermann Weyl and Eugene Wigner.The chief advantage of the phase-space formulation is that it makes quantum mechanics appear as similar to Hamiltonian mechanics as possible by avoiding the operator formalism, thereby "'freeing' the quantization of the 'burden' of the Hilbert space". This formulation is statistical in nature and offers logical connections between quantum mechanics and classical statistical mechanics, enabling a natural comparison between the two (see classical limit). Quantum mechanics in phase space is often favored in certain quantum optics applications (see optical phase space), or in the study of decoherence and a range of specialized technical problems, though otherwise the formalism is less commonly employed in practical situations.The conceptual ideas underlying the development of quantum mechanics in phase space have branched into mathematical offshoots such as algebraic deformation theory (see Kontsevich quantization formula) and noncommutative geometry.