Search
- https://phys.libretexts.org/Bookshelves/Astronomy__Cosmology/Supplemental_Modules_(Astronomy_and_Cosmology)/Cosmology/Carlip/Quantum_GeometryBut although it is a quantum theory based on general relativity, it is not entirely clear that it is really a "quantum theory of gravity" -- the time evolution remains only partially understood, and w...But although it is a quantum theory based on general relativity, it is not entirely clear that it is really a "quantum theory of gravity" -- the time evolution remains only partially understood, and we do not yet know how to recover a good classical limit that looks like classical general relativity.
- https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Quantum_Mechanics_(Fowler)These notes cover the essential core topics of a standard graduate quantum mechanics course. The first lectures here review undergraduate quantum and modern physics as a reminder, this optional materi...These notes cover the essential core topics of a standard graduate quantum mechanics course. The first lectures here review undergraduate quantum and modern physics as a reminder, this optional material was only covered briefly in class.
- https://phys.libretexts.org/Bookshelves/College_Physics/College_Physics_1e_(OpenStax)/30%3A_Atomic_Physics/30.08%3A_Quantum_Numbers_and_Ruleshysical characteristics that are quantized -- such as energy, charge, and angular momentum -- are of such importance that names and symbols are given to them. The values of quantized entities are expr...hysical characteristics that are quantized -- such as energy, charge, and angular momentum -- are of such importance that names and symbols are given to them. The values of quantized entities are expressed in terms of quantum numbers , and the rules governing them are of the utmost importance in determining what nature is and does. This section covers some of the more important quantum numbers and rules.
- https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Quantum_Mechanics_III_(Chong)/05%3A_Quantum_ElectrodynamicsThis chapter gives an introduction to quantum electrodynamics, the quantum theory of the electromagnetic field and its interactions with electrons and other charged particles.
- https://phys.libretexts.org/Courses/Prince_Georges_Community_College/General_Physics_I%3A_Classical_Mechanics/63%3A_Quantum_MechanicsAlthough classical mechanics can be applied to a wide range of situations, it was discovered at the beginning of the 20th century that it cannot be applied to very small distance scales—say on the ord...Although classical mechanics can be applied to a wide range of situations, it was discovered at the beginning of the 20th century that it cannot be applied to very small distance scales—say on the order of the size of an atom or smaller. For these small distance scales, classical mechanics no longer works, and a completely different system of mechanics is needed, called quantum mechanics.
- https://phys.libretexts.org/Courses/Prince_Georges_Community_College/General_Physics_I%3A_Classical_Mechanics/63%3A_Quantum_Mechanics/63.02%3A_Quantum_MechanicsYou may wonder: how can it be that a particle is in many places at once, or that the place where it appears is completely unpredictable, or that it is in an unknown state unless we're measuring it, or...You may wonder: how can it be that a particle is in many places at once, or that the place where it appears is completely unpredictable, or that it is in an unknown state unless we're measuring it, or that it can be both a particle and a wave at the same time?
- https://phys.libretexts.org/Bookshelves/University_Physics/University_Physics_(OpenStax)/University_Physics_III_-_Optics_and_Modern_Physics_(OpenStax)/07%3A_Quantum_Mechanics/7.05%3A_The_Quantum_Particle_in_a_BoxIn this section, we apply Schrӧdinger’s equation to a particle bound to a one-dimensional box. This special case provides lessons for understanding quantum mechanics in more complex systems. The energ...In this section, we apply Schrӧdinger’s equation to a particle bound to a one-dimensional box. This special case provides lessons for understanding quantum mechanics in more complex systems. The energy of the particle is quantized as a consequence of a standing wave condition inside the box.
- https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Quantum_Mechanics_III_(Chong)/03%3A_Quantum_Entanglement/3.05%3A_Quantum_CryptogaphyAccording to Equation \(\eqref{clone}\), \[\begin{align}\begin{aligned} \Big(\langle \psi_1 | \langle 0 | \hat{U}^\dagger \Big) \Big(\hat{U} | \psi_2 \rangle |0\rangle \Big) &= \Big(\langle \psi_1| \l...According to Equation \(\eqref{clone}\), \[\begin{align}\begin{aligned} \Big(\langle \psi_1 | \langle 0 | \hat{U}^\dagger \Big) \Big(\hat{U} | \psi_2 \rangle |0\rangle \Big) &= \Big(\langle \psi_1| \langle \psi_1| e^{-i\phi_1} \Big) \Big( e^{i\phi_2} |\psi_2\rangle|\psi_2\rangle\Big) \\ &= e^{-i(\phi_1-\phi_2)} \Big( \langle\psi_1 | \psi_2\rangle \Big)^2. \end{aligned}\label{clone1}\end{align}\] Here, \(\phi_1\) and \(\phi_2\) are the phase factors from Equation \(\eqref{clone}\) for the two ch…
- https://phys.libretexts.org/Courses/Muhlenberg_College/MC_%3A_Physics_213_-_Modern_Physics/04%3A_Quantum_Mechanics/4.05%3A_The_Quantum_Particle_in_a_BoxIn this section, we apply Schrӧdinger’s equation to a particle bound to a one-dimensional box. This special case provides lessons for understanding quantum mechanics in more complex systems. The energ...In this section, we apply Schrӧdinger’s equation to a particle bound to a one-dimensional box. This special case provides lessons for understanding quantum mechanics in more complex systems. The energy of the particle is quantized as a consequence of a standing wave condition inside the box.
- https://phys.libretexts.org/Courses/Bowdoin_College/Phys1140%3A_Introductory_Physics_II%3A_Part_2/06%3A_Quantum_Mechanics/6.05%3A_The_Quantum_Particle_in_a_BoxIn this section, we apply Schrӧdinger’s equation to a particle bound to a one-dimensional box. This special case provides lessons for understanding quantum mechanics in more complex systems. The energ...In this section, we apply Schrӧdinger’s equation to a particle bound to a one-dimensional box. This special case provides lessons for understanding quantum mechanics in more complex systems. The energy of the particle is quantized as a consequence of a standing wave condition inside the box.
- https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Quantum_Mechanics_III_(Chong)/04%3A_Identical_Particles/4.04%3A_Quantum_Field_TheoryFor simplicity, suppose the particles are bosons, and let \[\hat{\psi}(\mathbf{r}) = \sum_\mu \varphi_\mu(\mathbf{r}) \, \hat{a}_\mu, \quad\;\; \hat{\psi}^\dagger(\mathbf{r}) = \sum_\mu \varphi_\mu^*(...For simplicity, suppose the particles are bosons, and let \[\hat{\psi}(\mathbf{r}) = \sum_\mu \varphi_\mu(\mathbf{r}) \, \hat{a}_\mu, \quad\;\; \hat{\psi}^\dagger(\mathbf{r}) = \sum_\mu \varphi_\mu^*(\mathbf{r}) \, \hat{a}_\mu^\dagger.\] Using the aforementioned wavefunction properties, we can derive the inverse relations \[\hat{a}_\mu = \int d^dr \; \varphi_\mu^*(\mathbf{r}) \, \hat{\psi}(\mathbf{r}), \quad\;\; \hat{a}_\mu^\dagger = \int d^dr \; \varphi_\mu(\mathbf{r}) \, \hat{\psi}^\dagger(\m…
