13 Mechanical Waves Fall 2003
... Simultaneously, the portion on the left exerts a transverse force on the portion to the right, given by the negative of this expression (according to Newton’s third law). Reflection, Superposition, and Standing Waves Suppose a wave pulse is initiated at the positive-x end of a stretched rope and tra ...
... Simultaneously, the portion on the left exerts a transverse force on the portion to the right, given by the negative of this expression (according to Newton’s third law). Reflection, Superposition, and Standing Waves Suppose a wave pulse is initiated at the positive-x end of a stretched rope and tra ...
Quantum Mechanical
... has only certain _________________. • Bohr suggested that the single electron in a hydrogen atom moves around the nucleus in only certain allowed circular orbits. ...
... has only certain _________________. • Bohr suggested that the single electron in a hydrogen atom moves around the nucleus in only certain allowed circular orbits. ...
Sections 4 - Columbia Physics
... 3. A thin plate with surface rest-mass density Σ0 [g/cm2 ] is surrounded by uniform dust at rest with mass density ρ [g/cm3 ]. At time t = 0 the plate is set in motion along its normal with initial Lorentz factor γ(0) = γ0 . The moving plate collides inelastically with the dust particles (which sti ...
... 3. A thin plate with surface rest-mass density Σ0 [g/cm2 ] is surrounded by uniform dust at rest with mass density ρ [g/cm3 ]. At time t = 0 the plate is set in motion along its normal with initial Lorentz factor γ(0) = γ0 . The moving plate collides inelastically with the dust particles (which sti ...
Lagrangians and Local Gauge Invariance
... The coefficients like 0=1 and 1= 2= 3=i do not work since they do not eliminate the cross terms. It would work if these coefficients are matrices that satisfy the conditions ...
... The coefficients like 0=1 and 1= 2= 3=i do not work since they do not eliminate the cross terms. It would work if these coefficients are matrices that satisfy the conditions ...
Wick Rotation as a New Symmetry
... up with the classical path which is represented by the thick line in figure 1(b). The particle will progress along a single path that has maximum probability. Thus the difference between the two approaches can be traced to the state of crystallization of reality. This leads us to conclude that the i ...
... up with the classical path which is represented by the thick line in figure 1(b). The particle will progress along a single path that has maximum probability. Thus the difference between the two approaches can be traced to the state of crystallization of reality. This leads us to conclude that the i ...
- Philsci
... of a transaction occurs with respect to the endpoints of a space-time interval or intervals, rather than at a particular instant of time, the latter being a non-covariant notion. Its compatibility with relativity is also evident in that it makes use of both the positive and negative energy solutions ...
... of a transaction occurs with respect to the endpoints of a space-time interval or intervals, rather than at a particular instant of time, the latter being a non-covariant notion. Its compatibility with relativity is also evident in that it makes use of both the positive and negative energy solutions ...
Energy Levels Of Hydrogen Atom Using Ladder Operators
... This time the energy is !ω more than that of the ground state. When † is applied again, the energy will increase by a further !ω, ie two units of energy above the ground state, and so on. In this context, † and  are known as the ladder operators. The figure below [1] indicates the raising and ...
... This time the energy is !ω more than that of the ground state. When † is applied again, the energy will increase by a further !ω, ie two units of energy above the ground state, and so on. In this context, † and  are known as the ladder operators. The figure below [1] indicates the raising and ...
Why is Quantum Science Disturbing
... at the atomic level of nature. In quantum physics, there appears to be an eerie connection between the physical state of a system and conscious awareness of it by some observing being. But it's really the act of measurement by any other system that resets, or "collapses," the quantum state into one ...
... at the atomic level of nature. In quantum physics, there appears to be an eerie connection between the physical state of a system and conscious awareness of it by some observing being. But it's really the act of measurement by any other system that resets, or "collapses," the quantum state into one ...
Observables and Measurements in Quantum Mechanics
... cavity field. Photons are typically counted by photodetectors whose mode of operation is to absorb a photon and create a pulse of current. So we may well be able to count the number of photons in the field, but in doing so, there is no field left behind after the counting is completed. All that we c ...
... cavity field. Photons are typically counted by photodetectors whose mode of operation is to absorb a photon and create a pulse of current. So we may well be able to count the number of photons in the field, but in doing so, there is no field left behind after the counting is completed. All that we c ...
Solving the Time-Independent Schrödinger Equation Abstract
... Let’s start by looking for an even solution. Begin at the middle of the box and choose a value for ψ(0). (It doesn’t matter what value we choose here – we must eventually normalize the wavefunction anyway.) An even solution will also have ψ 0 (0) ≡ dψ/dx|x=0 = 0. Now, the TISE is a second-order diff ...
... Let’s start by looking for an even solution. Begin at the middle of the box and choose a value for ψ(0). (It doesn’t matter what value we choose here – we must eventually normalize the wavefunction anyway.) An even solution will also have ψ 0 (0) ≡ dψ/dx|x=0 = 0. Now, the TISE is a second-order diff ...
classical and
... the set of equations given in expression (12).” As occur in any system of differential equations, solving (13) will give us a very general solution containing parameters to be determined by the initial conditions, i.e. position and velocity at a specific time t = to. Once the initial conditions are ...
... the set of equations given in expression (12).” As occur in any system of differential equations, solving (13) will give us a very general solution containing parameters to be determined by the initial conditions, i.e. position and velocity at a specific time t = to. Once the initial conditions are ...
PHYS201 - Wave Mechanics
... frequency f in multiples of hf where h was a parameter to be taken to zero at the end of the calculation. But he got the correct answer for a small but non-zero value of h: ...
... frequency f in multiples of hf where h was a parameter to be taken to zero at the end of the calculation. But he got the correct answer for a small but non-zero value of h: ...
8.514 Many-body phenomena in condensed matter and atomic
... As a simple application of the operators a and a+ let us reconstruct the main facts of the harmonic oscillator quantum mechanics. 1. The ground state | ψ0〉also called vacuum state, provides the lowest possible energy ...
... As a simple application of the operators a and a+ let us reconstruct the main facts of the harmonic oscillator quantum mechanics. 1. The ground state | ψ0〉also called vacuum state, provides the lowest possible energy ...
Wave function
A wave function in quantum mechanics describes the quantum state of an isolated system of one or more particles. There is one wave function containing all the information about the entire system, not a separate wave function for each particle in the system. Its interpretation is that of a probability amplitude. Quantities associated with measurements, such as the average momentum of a particle, can be derived from the wave function. It is a central entity in quantum mechanics and is important in all modern theories, like quantum field theory incorporating quantum mechanics, while its interpretation may differ. The most common symbols for a wave function are the Greek letters ψ or Ψ (lower-case and capital psi).For a given system, once a representation corresponding to a maximal set of commuting observables and a suitable coordinate system is chosen, the wave function is a complex-valued function of the system's degrees of freedom corresponding to the chosen representation and coordinate system, continuous as well as discrete. Such a set of observables, by a postulate of quantum mechanics, are Hermitian linear operators on the space of states representing a set of physical observables, like position, momentum and spin that can, in principle, be simultaneously measured with arbitrary precision. Wave functions can be added together and multiplied by complex numbers to form new wave functions, and hence are elements of a vector space. This is the superposition principle of quantum mechanics. This vector space is endowed with an inner product such that it is a complete metric topological space with respect to the metric induced by the inner product. In this way the set of wave functions for a system form a function space that is a Hilbert space. The inner product is a measure of the overlap between physical states and is used in the foundational probabilistic interpretation of quantum mechanics, the Born rule, relating transition probabilities to inner products. The actual space depends on the system's degrees of freedom (hence on the chosen representation and coordinate system) and the exact form of the Hamiltonian entering the equation governing the dynamical behavior. In the non-relativistic case, disregarding spin, this is the Schrödinger equation.The Schrödinger equation determines the allowed wave functions for the system and how they evolve over time. A wave function behaves qualitatively like other waves, such as water waves or waves on a string, because the Schrödinger equation is mathematically a type of wave equation. This explains the name ""wave function"", and gives rise to wave–particle duality. The wave of the wave function, however, is not a wave in physical space; it is a wave in an abstract mathematical ""space"", and in this respect it differs fundamentally from water waves or waves on a string.For a given system, the choice of which relevant degrees of freedom to use are not unique, and correspondingly the domain of the wave function is not unique. It may be taken to be a function of all the position coordinates of the particles over position space, or the momenta of all the particles over momentum space, the two are related by a Fourier transform. These descriptions are the most important, but they are not the only possibilities. Just like in classical mechanics, canonical transformations may be used in the description of a quantum system. Some particles, like electrons and photons, have nonzero spin, and the wave function must include this fundamental property as an intrinsic discrete degree of freedom. In general, for a particle with half-integer spin the wave function is a spinor, for a particle with integer spin the wave function is a tensor. Particles with spin zero are called scalar particles, those with spin 1 vector particles, and more generally for higher integer spin, tensor particles. The terminology derives from how the wave functions transform under a rotation of the coordinate system. No elementary particle with spin 3⁄2 or higher is known, except for the hypothesized spin 2 graviton. Other discrete variables can be included, such as isospin. When a system has internal degrees of freedom, the wave function at each point in the continuous degrees of freedom (e.g. a point in space) assigns a complex number for each possible value of the discrete degrees of freedom (e.g. z-component of spin). These values are often displayed in a column matrix (e.g. a 2 × 1 column vector for a non-relativistic electron with spin 1⁄2).In the Copenhagen interpretation, an interpretation of quantum mechanics, the squared modulus of the wave function, |ψ|2, is a real number interpreted as the probability density of measuring a particle as being at a given place at a given time or having a definite momentum, and possibly having definite values for discrete degrees of freedom. The integral of this quantity, over all the system's degrees of freedom, must be 1 in accordance with the probability interpretation, this general requirement a wave function must satisfy is called the normalization condition. Since the wave function is complex valued, only its relative phase and relative magnitude can be measured. Its value does not in isolation tell anything about the magnitudes or directions of measurable observables; one has to apply quantum operators, whose eigenvalues correspond to sets of possible results of measurements, to the wave function ψ and calculate the statistical distributions for measurable quantities.The unit of measurement for ψ depends on the system, and can be found by dimensional analysis of the normalization condition for the system. For one particle in three dimensions, its units are [length]−3/2, because an integral of |ψ|2 over a region of three-dimensional space is a dimensionless probability.