Simply Lecture

Recall the relationship: $$\phi(t) = C_+ exp[-i(\frac{E}{\hbar})t] + C_- exp[i(\frac{E}{\hbar})t]$$ .............(1) Recall, from the present special case: $$C_+ = 0$$ ........(2) Substitute equ (2) into equ (1), we have that: $$\phi(t) = C_- exp[i(\frac{E}{\hbar})t]$$ .......(3) Also,  recall the relationship: $$\Psi(r,t) = \psi(r) \phi(t)$$ ..........(4) Substitute equ (3) into equ (4): $$\Psi(r,t) = C_- exp \psi (r)  [i(\frac{E}{\hbar})t]$$ ..............(5) Equ (5) can be differentiated with respect to t, and by rearranging we have: $$\Psi(r,t) = -i(\frac{\hbar}{E}) \frac{d \Psi (r, t)}{dt}$$ ..............(6) Recall the relationship: $$-\frac{\hbar^{2}}{2m} \nabla^{2} \Psi(r,t) + V(r) \Psi(r,t) = E \Psi(r,t)$$ ...........(7) Substituting equ (6) into the right hand side of equ (7) we have: $$-\frac{\hbar^{2}}{2m} \nabla^{2} \Psi(r,t) + V(r) \Psi(r,t) = i\hbar \frac{d \Psi(r,t)}{dt}$$ .........(8) Equ (8) is the auxiliary time dependent Schrodinger equation...

Recall the relationship: $$k = \frac{E}{\hbar}$$ ........(1) And also the relationship: $$(\nabla^{2} + k^{2}) \psi (r)=0$$ ........(2) Substitute equ (1) into equ (2) $$(\nabla^{2} + \frac{E^{2}}{\hbar^{2}}) \psi (r) = 0$$ Opening the brackets: $$\nabla^{2} \psi(r) + \frac{E^{2}}{\hbar^{2}} \psi(r) =0$$ Therefore: $$\nabla^{2} \psi(r) = -(\frac{E}{\hbar})^{2} \psi(r)$$ ............(3) Multiply both sides of equ (3) by \(-\frac{\hbar^{2}}{2m}\), we have: $$-\frac{\hbar^{2}}{2m} \nabla^{2} \psi(r) = \frac{\hbar^{2}}{2m} (\frac{E^{2}}{\hbar^{2}}) \psi(r)$$ $$-\frac{\hbar^{2}}{2m} \nabla^{2} \psi(r) = (\frac{E^{2}}{2m}) \psi(r)$$ ..........(4) Recall, the relationship from the energy form of the Schrodinger equation, we have: $$\frac{E^{2}}{2m} = E - V(r)$$ .............(5) Substitute equ (5) into equ (4): $$-\frac{\hbar^{2}}{2m} \nabla^{2} \psi(r) = (E - V(r)) \psi(r)$$ Opening bracket,  we have: $$-\frac{\hbar^{2}}{2m} \nabla^2 \psi (r) = E \psi (r) - V(r) \psi (r)...

Recall from the time independent Schrodinger equation: $$-\frac{\hbar^{2}}{2m} \nabla^{2} \psi(r) + V(r) \psi(r) = E \psi(r)$$ ........(1) Multiply both sides of equ (1) by \(\phi(t)\), we have: $$-\frac{\hbar^{2}}{2m} \nabla^{2} \psi(r) \phi(t) + V(r) \psi(r) \phi(t) = E \psi(r) \phi(t)$$ ..........(2) Recall: $$\psi(r) \phi(t) = \Psi(r,t)$$ .............(3) Substitute equ (3) into equ (2): $$-\frac{\hbar^{2}}{2m} \nabla^{2} \Psi(r,t) + V(r) \Psi(r,t) = E \Psi(r,t)$$ .........(4) Recall that the well known Planck-Einstein relation is given by: $$E = hv$$ ...........(5) And also, the reduced Planck's constant is given by: $$\hbar = \frac{h}{2\pi}$$ ................(6) Also recall that time dependence is given by: $$\phi(t) = C_+ exp(-i2\pi vt) + C_- exp(i2\pi vt)$$ ..........(7) Substituting equ (6) into (5), we have: $$V = \frac{E}{\hbar 2 \pi}$$ ..........(8) Substitute equ (8) into equ (7) $$\phi(t) = C_+ exp(-i2 \pi \frac{E}{\hbar 2 \pi} t)  + C_- exp(i2 \pi \f...

Recall that classical energy is given by the equation: $$E_c = \frac{p^{2}}{2m} + V(r) + V_0$$ ........(1) Where, V( r ) = spatially dependent potential              \(V_0\) = constant potential energy Recall from the square of the magnitude of the momenergy 4-vector associated with quantum object is given by: $$E^{2} - p^{2} = m^{2}$$ ..........(2) Divide equ (2) by 2m $$\frac{E^{2}}{2m} - \frac{p^{2}}{2m} = \frac{m^{2}}{2m}$$ $$\frac{E^{2}}{2m} - \frac{p^{2}}{2m} = \frac{m}{2}$$ Making \(\frac{p^{2}}{2m}\) subject of the formula, we have: $$\frac{p^{2}}{2m} = \frac{E^{2}}{2m} - \frac{m}{2}$$ .........(3) Also let: $$V_0 = \frac{m}{2} + E_c - E$$ ............(4) Substitute equ (4) and equ (3) into equ (1), we have: $$E_c = \frac{E^{2}}{2m} - \frac{m}{2} + V(r) + \frac{m}{2} + E_c - E$$ Making \(\frac{E^{2}}{2m}\) subject of the formula, we have: $$\frac{E^{2}}{2m} = E_c - E_c + \frac{m}{2} - \frac{m}{2} - V(r) + E$$ $$\frac{E^{2}}{2m}...

Any vibration that disturbs air molecule, is capable of making sound. The source of vibration pushes and pull air molecules, causing acoustic signals known as sounds. The sound heard are based on three major factors; that is: The source of vibration that forms the sound waves. The medium which is known as the wave carrier. The receiver that detect the sound. The source of vibration occilates and brings surrounding air into motion through a medium (such as air) and if the receiver is present the sound can be perceived. Sound is a mechanical disturbance that travels through an elastic or viscous medium at a speed depending on the characteristics or features of the medium. While air tubulates, the mass and momentum of the sound are the "sound generators". The diagram below shows how the sound are generated. Diagram showing how sounds are generated.

Sound wave propagation is a wave process. The acoustic signals requires a mechanical elastic medium for propagation and cannot travel through a vacuum unlike, electromagnetic light waves. Sounds travels more quickly in solids, followed by liquid than through gases. The velocity of sound, in a wave form is 331.29 {ms}^-1 through dry air at {0}^oC Sound wave concept When vibration disturbs particle in a medium, then the particle displaces other surrounding particles in that medium. A wave pattern is then formed, when the particle moves in the outward direction continuously. The wave carries the sound energy through the medium from the source and become less intense. The sound energy solely depends on the volume of the volume of the sound; the higher the sound energy the louder the volume. There are three concept of sound wave that causes a sound to be produced. They include: frequency (f), wavelength  \lambda and amplitude (A). The sound wave vibrates at different frequenc...

Maxwell equations are a set of four complicated equations that the world of electromagnetism. These equations shows how magnetic fields interacts and propagates. The Maxwell equation have rules, the universe use to govern the behavior of electric or magnetic field. The Maxwell equation was proposed by James Clerk Maxwell (1831-1879). The Maxwell equations are in both differential and integral forms.  Differential form: 1. $$\nabla.D = \rho V$$ 2. $$\nabla.B = 0$$ 3. $$\nabla X E = - \frac{dB}{dt}$$ 4. $$\nabla X  H = \frac{dD}{dt} + J$$ Integral form: 1. $$\int(D.ds) = \int_{v} \rho dV$$ 2. $$\psi_m = \int B.ds = 0$$ 3. $$\int E.dL = -\int_{s} \frac{dB.ds}{dt}$$ 4.$$\int H.dL = \int (J + \frac{dD}{dt}).ds$$ To find out about the derivation of Maxwell equations, click the links below: Maxwell first equation Maxwell second equation   Maxwell third equation Maxwell fourth equation