Chyi-Lung Lin

• Soochow University

We show the method for constructing nonspreading wave packets whose shape and motion can be general. We analyze the time evolution of nonspreading wave packets by decomposing the Hamiltonian into two parts. Of the two, one changes the instantaneous state, the other does not. Through this decomposition, the time evolution operator is shown to be eff...

For a quantum confinement model, the wave function of a particle is zero outside the confined region. Due to this, the negative energy states are, in fact, square integrable. As negative energy states are not physical, we need to impose some boundary conditions in order to avoid these states. For the case of the infinite square well model, we show...

Ehrenfest's theorem in the infinite square well is up to now only manifested indirectly. The manifestation of this theorem is first done in the finite square well, and then consider the infinite square well as the limit of the finite well. For a direct manifestation, we need a more precise formula to describe the degree of infiniteness of the diver...

The concept of photon is not necessary only applied to the relativistic Doppler theory. It may also work well for classical theory. As conservation of momentum and energy are physical laws, if applying these laws gives the exact relativistic Doppler effect, it should also give the exact classical Doppler effect. So far the classical Doppler effect...

The original model of the infinite square well contains a vague notation infinity and therefore results some ambiguities. We investigate to obtain a functional form for the potential energy V(x). This is done by substituting back the original energy eigenstates and eigenvalues into the Schrodinger equation. We then obtain a precise functional form...

We show that it needs a more delicate potential to confine particles inside a well. The original model containing a vague notation ∞ in the potential energy is ambiguous. Using the Heaviside step function θ(x) and the Dirac delta-function δ(x), we give a precise form for the confining potential. Although such form appears unusual, the ambiguities a...

We show that it needs a more delicate potential to confine particles inside a well. The original model containing a vague notation of infinity in the potential energy is ambiguous. Using the Heaviside step function and the Dirac delta-function, we give a precise form for the confining potential. Although such form appears unusual, the ambiguities a...

For analyzing the time evolution of quantum mechanically driven harmonic oscillator systems, we first establish the eigenvalue equation for the time-evolved wave function, and then we dynamically decompose the Hamiltonian into two parts. One part is the operator which does not change the state, and the other part does. Through this decomposition, t...

We analyze the propagation dynamics of nonspreading wave packets by decomposing the Hamiltonian H into two parts: H = (H) over tilde (t) + H-c(t). The first part (H) over tilde (t) is such that Psi(x, t) is its instantaneous eigenstate, and is therefore irrelevant to the propagation of the packet. The second part H-c(t) is shown to be the effective...

We show a new method for analyzing the time evolution of the Schrodinger wave function phi(x,t). We propose the decomposition of the Hamiltonian as: H(t)=Hp(t)+Hc(t), where Hp(t)is the operator which does not change the state and therefore phi(x,t) is its eigenfunction, and Hc(t)is the operator that changes the state. With this decomposition, the t...

We show a new method for analyzing the time evolution of the Schrodinger wave function Psi(x,t). We propose the decomposition of the Hamiltonian as: H(t)=Hp(t)+Hc(t), where Hp(t) is the Hamiltonian such that Psi(x,t) is its instantaneous eigenfunction, and Hc(t) the Hamiltonian which changes the state Psi. With this decomposition, the action of H(t...

We discuss the propagation dynamics of nonspreading wave packets. We decompose the Hamiltonian into two parts. The first part is such that wave packets is its instantaneous eigenstate and is therefore irrelevant to the propagation of the packet. The second part is shown to be the effective Hamiltonian governing the motion of the packet both classic...

Berry and Balazs showed that an initial Airy packet Ai(b x) under time evolution is nonspreading in free space and also in a homogeneous time-varying linear potential V(x,t)=-F(t) x. We find both results can be derived from the time evolution operator U(t). We show that U(t) can be decomposed into ordered product of operators and is essentially a s...

We start from the simple fact that the method of images can always be used to obtain {\Phi}_{\sigma} (r^\to_in), which is the potential inside conductor produced by induced surface charges. We use this fact to construct image method for outside potential {\Phi}_{\sigma} (r^\to_out). We show that if we can find a relation between {\Phi}_{\sigma} (r^...

We discuss nonspreading wave packets in one dimensional Schr\"{o}dinger equation. We derive general rules for constructing nonspreading wave packets from a general potential $\textmd{V}(x,t)$. The essential ingredients of a nonspreading wave packet, the shape function $f(x)$, the motion $d(t)$, the phase function $\phi(x,t)$ are derived. Since the...

We show that for Hamiltonian of the form, H = c(t)∂x∂x +V (x, t), the general potential V (x, t) for which Airy packets remain nonspreading under time evolution can only be of the form V (x, t)= p(t)x+q(t), where p(t) and q(t) are real functions. We derive rules for constructing nonspreading Airy packets from this general potential. We also conside...

We analyze the light path in a spherically symmetrical medium, n(r) = root1 + r(0)(2)/r(2). We find that the ray path may be viewed as a trajectory of a particle for which the mass is velocity dependent. It is found to be m = m(o)n(2) = m(o)c(2)/v(2), where m(o) is an arbitrary constant mass and v is the speed of light in the medium. The equation g...

Based on reversing Hamilton's principle, the Lagrangian can be obtained directly from the equations of motion. The method is illustrated for a relativistic particle moving in an external field, the electromagnetic field, and a damped harmonic oscillator.

We examine total reflection for waves propagating from an isotropic medium to an anisotropic medium. By calculating the value, ni=neff (µt), the ratio of the indices of refrac- tion of an isotropic medium and an anisotropic medium, it is found that total reflection can occur for waves propagating from a rarer medium to a denser medium. This is cont...

We show that interesting dividing formulas such as, Chinese theorem, Fermat's little theorem, and Euler's theorem can easily be derived from some well-known iterated maps. Other dividing formulas concerning Fibonacci numbers, generalized Fibonacci numbers of degree m, and numbers of other types can also be derived. The results show that iterated ma...

We examine the value n i / n eff θ t , the ratio of the indices of refraction for waves transmitted from an isotropic medium to an anisotropic medium. It is found that total reflection can occur in the case of propagation from a rarer medium to a denser medium.