It can be shown that, for continuity of u and u' at xl, k = 27r/8 = 45 deg. The surface through xl, serving as a boundary between the regions occupied by the exponential and oscillatory solutions, is the well-known potential barrier. There is an equivalence between much of the W-K-B method and the following simpler set of equations. In place of the time-eliminated wave equation of optics, write (d2u/dx2) sin2a + (I/x) (du/dx) cos2a + (47r2/X2)U = O, which becomes the time-eliminated wave equation of optics in one dimension when a = 90 deg. Solutions: u' -A exp (-27r i tanr 0.34657),. . . x , ... where cosa = I/x, sina (x2-X2) 1/2/x, tana = (x2-X2) 1//X . When x = -, u'= 0.7071 A =A sin 27r/8 = u. It may be shown that du'/dx = d'u/dx5 when x = ?+ . The surface through x = -X, serving as a boundary between the regions occupied by the exponential and oscillatory solutions, is, therefore, equivalent to the W-K-B potential barrier. In fact, the above simple equations may be given the form of the W-K-B equations by the same substitutions which transform the wave equation of optics into the Schr6dinger equation. When a consistent vector system of directions is assigned to u, x and X, the wave motion, described by the above simple equations, has conical wave fronts.
It can be shown that, for continuity of u and u' at xl, k = 27r/8 = 45 deg. The surface through xl, serving as a boundary between the regions occupied by the exponential and oscillatory solutions, is the well-known potential barrier. There is an equivalence between much of the W-K-B method and the following simpler set of equations. In place of the time-eliminated wave equation of optics, write (d2u/dx2) sin2a + (I/x) (du/dx) cos2a + (47r2/X2)U = O, which becomes the time-eliminated wave equation of optics in one dimension when a = 90 deg. Solutions: u' -A exp (-27r i tanr 0.34657),. . . x , ... where cosa = I/x, sina (x2-X2) 1/2/x, tana = (x2-X2) 1//X . When x = -, u'= 0.7071 A =A sin 27r/8 = u. It may be shown that du'/dx = d'u/dx5 when x = ?+ . The surface through x = -X, serving as a boundary between the regions occupied by the exponential and oscillatory solutions, is, therefore, equivalent to the W-K-B potential barrier. In fact, the above simple equations may be given the form of the W-K-B equations by the same substitutions which transform the wave equation of optics into the Schr6dinger equation. When a consistent vector system of directions is assigned to u, x and X, the wave motion, described by the above simple equations, has conical wave fronts.
A wave equation already exists in classical physics (for example for a vibrating string) with a solution of the form exp(ikx-iwt) where k=2*3.14/wavelength and w=2*3.14 frequency. K and w are not simply linked to momentum and energy, but the wave is physically visible. The wavelike function exp(ikx-iEt) also appears in solutions for the photon, the relativistic Klein Gordon equation (and the Dirac too) as well as in the Schrodinger equation. This solution separates x and k and E and t so -id/dx exp(ikx-iEt) = k exp(ikx-iEt) and id/dt exp(ikx-iEt) = E exp(ikx-iEt). If a classical equation exists linking energy and momentum (like EE=pp+momo or E=.5mvv), one may convert it into a differential equation with independent fluctuations in x and t, but linked through the energy relationship. This retains the classical information, but introduces a wave solution. For the Schrodinger equation with a potential, exp(-iEt + i Integral (0,x) dx1 v(x1)m is a solution which is equivalent to exp(i Integral (0,t) L dt1) where L is the Lagrangian. In this case, -id/dx exp(i action) = p(x) exp(i action) and id/dt exp(i action) = E exp(i action). This also holds for the relativistic Lagrangian -mo sqrt(1-vv). In the case of an electromagnetic potential v dot A, one may write Integral dt1 v dot A = Integral (0,x) dx1 dot A and so A behaves exactly like p(x). For the case of light, exp(ikx-iEt) follows as a solution of a wave equation for either the electric or magnetic fields (which follows from Maxwell’s equations). The equation: d/dt (.5eoEl El + .5/uo BB) + grad dot El x B = 0 (for no source) also follows from Maxwell’s equations. It is an equation relating energy density to momentum flux and also allows for an exp(ikx-iEt) solution if El and B have the same magnitude (c=1) and are perpendicular to each other and the motion of the photon.
A potential barrier of the kind studied by Fowler and others may be represented by the analytic function V (Eq. (1)). The Schr\"odinger equation associated to this potential is soluble in terms of hypergeometric functions, and the coefficient of reflection for electrons approaching the barrier with energy W is calculable (Eq. (15)). The approximate formula, $1\ensuremath{-}\ensuremath{\rho}=\mathrm{exp}{\ensuremath{-}\ensuremath{\int}\frac{4\ensuremath{\pi}}{h}{(2m(V\ensuremath{-}W))}^{\frac{1}{2}}\mathrm{dx}}$ is shown to agree very well with the exact formula when the width of the barrier is great compared to the de Broglie wave-length of the incident electron, and W<Vmax.
In (1), an outline of the derivation of the Gamow factor for alpha decay from a nucleus is given. In one dimension, the approach considers three regions in space, one with an incident and reflected free particle wavefunction, the second (within a square V barrier) with an exponentially growing and falling solution, and the third, with a single free particle wavefunction exp(-iEt)exp(ipx+phase) representing the escaped alpha particle. Alpha decay is treated as a quantum tunneling problem and the wavefunction and its first derivative d/dx must be continuous. A second step in (1), however, is to assume that one may write a time related wavefunction as exp(- g t), where d/dt partial Integral dx (region) W*W = ½ (-i W* d/dx W + i W d/dx W*). This may appear to be a quantum mechanical equation, but we argue that this is nothing more than the classical continuity equation. In fact, in (2) we argued that quantum mechanics does not exist without classical equations. In particular, the quantum free particle wavefunctions exp(-iEt) and exp(ipx) follow exp(iEt)exp(-iEt) = 1 and exp(-ipx)exp(ipx) = 1 to link them classical densities in time and space for particles moving with a constant speed. If there is a decay in time and space, then the appropriate classical equation is the continuity one. Writing d/dt partial d(t) = constant d(t) yields the familiar exp(-g t) form. We apply the ideas of the Gamow factor to a set of photons bouncing back and forth in an n1 1-dimensional cavity surrounded by n2 (n2>n1) index of refraction material. In this case, if the photons are synchronized then one may follow them classically. The catch is that the probability to reflect or refract is quantum. In general, one would not write exp(-g t) in such a scenario, but would formally write an equation of the form: fraction of remaining photons = R power Integer(t/t1), where R is the probability to reflect and integer(t/t1) is the number of cycles.. If, however, the probability to refract is very small, we show that one may in fact obtain the same result as in the case of exp(- g t) for g <<1. Alternatively, if the cavity is small in length and the speed of the particle (c in the case of light) is large, one could approximate the situation by d/dt partial d(t)= constant d(t) and then obtain exp(-g t), having the photons no longer in deterministic motion.
In the past few chapters, the Schrödinger equation has been developed and applied to the solution of a number of quantum mechanical problems, mainly to develop experience with the results of simple and common systems. The results from these examples can be summarized relatively simply. In general, quantization enters the problem through the non-commuting nature of conjugate operators such as position and momentum as well as from classical boundary quantization. This has led to the Schrödinger equation itself being the primary equation of motion for the wave function solution to the problem. In essence, duality is incredibly important and is the basic method by which the electron is treated as a wave, rather than as a particle, and the wave equation of interest is the Schrödinger equation. When boundary conditions are applied, either through potential barriers or through the form of the potential (as in the harmonic oscillator), the time-independent Schrödinger equation yields solutions that are often special functions. Examples of this are the sines and cosines in the rectangular-barrier case, the Airy functions in the triangular-well case, and the Hermite polynomials in the harmonic oscillator case.
In previous studies, by the methods of wave mechanics, of one-dimensional motion of particles in cases in which there are intervals in which the value of the potential energy function V(x) exceeds the value of the total energy E, attention has been confined to wave functions of the form $f(x, E)\mathrm{exp}(\ensuremath{-}\frac{2\ensuremath{\pi}\mathrm{iEt}}{h})$. In the present note wave packets are considered, instead of these trains of waves.The function V(x) is taken as follows: V(x),=0forx<0andforx>a,=V0>0for0<x<a. A wave function is set up which initially represents a wave packet moving toward the point x=0 from the left. The separation of the incident packet into a reflected packet and a transmitted packet is studied. It is found that the transmitted packet appears at the point x=a at about the time at which the incident packet reaches the point x=0, so that there is no appreciable delay in the transmission of the packet through the barrier.
ly R is the Klein operator ±exp[iπ(Ĥ − E0)] while in Schrodinger coordinate representation, first investigated by Yang, R is realised by ±P where P is the parity operator: P|x >= ±|x >, P−1 = P, P = 1, PxP−1 = −x. (7) The basic (anti-)commutation relation (1) and (3) together with their derived relation (4) will be referred to here as constituting the WH algebra which is in fact a parabose algebra for one degree of freedom. We shall assume here in after, without loss of generality, that c is positive, i.e. c =| c |> 0. Thus, in coordinate representation the generalized quantization a la Wigner requires that x = x, ∞− < x < ∞, p = −i d dx + ic 2x P, R = P. (8) Indeed, following Yang representation [6] we obtain the coordinate representation for the ladder operators as given by â± −→ a±c 2 = 1 √ 2 ( ± d dx ∓ c 2x P − x ) . (9) Yang’s wave mechanical description was further investigated in [7, 8]. The Wigner-Heisenberg Algebra in Quantum Mechanics http://dx.doi.org/10.5772/55994 479 4 Advances in Quantum Mechanics The present author have applied a super-realization so that R = Σ3 to illustrate the first application of our operator method to the cases of the Hamiltonian of an isotonic oscillator (harmonic plus a centripetal barrier) system. To obtain a super-realisation of the WH algebra, we introduce, in addition to the usual bosonic coordinates (x,−i d dx ), the fermionic ones b∓(= (b±)†) that commute with the bosonic set and are represented in terms of the usual Pauli matrices Σi, (i = 1, 2, 3). Indeed, expressing a ±( c 2 ) in the following respective factorised forms: a( c 2 ) = 1 √ 2 Σ1x (1/2)cΣ3 exp( 1 2 x)( d dx )exp( −1 2 x2)x−(1/2)cΣ3 (10) a−( c 2 ) = 1 √ 2 Σ1x (1/2)cΣ3 exp( −1 2 x)( −d dx )exp( 1 2 x2)x−(1/2)cΣ3 , (11) where these lader operators satisfy the algebra of Wigner-Heisenberg. From (1), (11) and (10) the Wigner Hamiltonian becomes H( c 2 ) = 1 2 [ a( c 2 ), a−( c 2 ) ] + = ( H−( c 2 − 1) 0 0 H+( c 2 − 1) = H−( c 2 ) ) , (12) where the even and odd sector Hamiltonians are respectively given by H−( c 2 − 1) = 1 2 { − d 2 dx2 + x + 1 x2 ( c 2 )( c 2 − 1) }