However, we still are left wondering what the equations are that can be applied to properly describe such motions and why the extra conditions are needed It turns out that a new kind of equation based on combining wave and particle properties needed to be developed to address such issues. These are the so-called Schrodinger equations to which we now turn our attention As I said earlier, no one has yet shown that the Schrodiger equation follows deductively from some more fundamental theory. That is, scientists did not derive this equation, they postulated it. Some idea of how the scientists of that era"dreamed up the Schrodinger equation can be had by examining the time and spatial dependence that characterizes so-called travelling waves. It should be noted that the people who worked on these problems knew a great deal about waves(e.g, sound waves and water waves) and the equations they obeyed. Moreover, they knew that waves could sometimes displ the characteristic of quantized wavelengths or frequencies(e. g, fundamentals and overtones in sound waves). They knew, for example, that waves in one dimension that are constrained at two points(e.g, a violin string held fixed at two ends )undergo oscillatory motion in space and time with characteristic frequencies and wavelengths. For example, the motion of the violin string just mentioned can be described as having an amplitude A(x, t) at a position x along its length at time t given by A(x, t)=A(x, o)cos(2T v t) whe IS frequency. The amplitude' s spatial dependence also has sinusoidal dependence given by
11 However, we still are left wondering what the equations are that can be applied to properly describe such motions and why the extra conditions are needed. It turns out that a new kind of equation based on combining wave and particle properties needed to be developed to address such issues. These are the so-called Schrödinger equations to which we now turn our attention. As I said earlier, no one has yet shown that the Schrödiger equation follows deductively from some more fundamental theory. That is, scientists did not derive this equation; they postulated it. Some idea of how the scientists of that era “dreamed up” the Schrödinger equation can be had by examining the time and spatial dependence that characterizes so-called travelling waves. It should be noted that the people who worked on these problems knew a great deal about waves (e.g., sound waves and water waves) and the equations they obeyed. Moreover, they knew that waves could sometimes display the characteristic of quantized wavelengths or frequencies (e.g., fundamentals and overtones in sound waves). They knew, for example, that waves in one dimension that are constrained at two points (e.g., a violin string held fixed at two ends) undergo oscillatory motion in space and time with characteristic frequencies and wavelengths. For example, the motion of the violin string just mentioned can be described as having an amplitude A(x,t) at a position x along its length at time t given by A(x,t) = A(x,o) cos(2p n t), where n is its oscillation frequency. The amplitude’s spatial dependence also has a sinusoidal dependence given by
A(x,0)=Asin(2πx) where 2 is the crest-to-crest length of the wave. Two examples of such waves in one dimension are shown in Fig. 1. 4 A(x,0) sin(2TX/L) Figure 1. 4. Fundamental and first overtone notes of a violin string s,the string is fixed atx=0 and at x=L, so the wavelengths belonging to the two waves shown are 2=2L and n=L. If the violin string were not clamped at x the waves could have any value of 7. However, because the string is attached at x=L he allowed wavelengths are quantized to obey
12 A(x,0) = A sin (2p x/l) where l is the crest-to-crest length of the wave. Two examples of such waves in one dimension are shown in Fig. 1. 4. Figure 1.4. Fundamental and first overtone notes of a violin string. In these cases, the string is fixed at x = 0 and at x = L, so the wavelengths belonging to the two waves shown are l = 2L and l = L. If the violin string were not clamped at x = L, the waves could have any value of l. However, because the string is attached at x = L, the allowed wavelengths are quantized to obey A(x,0) x sin(1px/L) sin(2px/L)
where n=1, 2, 3, 4,..The equation that such waves obey called the wave equation. A(x, t)/dt=c'd'An where c is the speed at which the wave travels. This speed depends on the composition of the material from which the violin string is made. Using the earlier expressions for the x and t-dependences of the wave A(x, t), we find that the wave's frequency and wavelength are related by the so-called dispersion equation (c/) This relationship implies, for example, that an instrument string made of a very stiff material (large c)will produce a higher frequency tone for a given wavelength(i.e,a given value of n) than will a string made of a softer material(smaller c) For waves moving on the surface of, for example, a rectangular two-dimensional surface of lengths L- and L. one finds
13 l = L/n, where n = 1, 2, 3, 4, ... .The equation that such waves obey, called the wave equation, reads: d 2A(x,t)/dt2 = c2 d2A/dx2 where c is the speed at which the wave travels. This speed depends on the composition of the material from which the violin string is made. Using the earlier expressions for the xand t- dependences of the wave A(x,t), we find that the wave’s frequency and wavelength are related by the so-called dispersion equation: n 2 = (c/l) 2 , or c = l n. This relationship implies, for example, that an instrument string made of a very stiff material (large c) will produce a higher frequency tone for a given wavelength (i.e., a given value of n) than will a string made of a softer material (smaller c). For waves moving on the surface of, for example, a rectangular two-dimensional surface of lengths Lx and Ly , one finds
A(x,y, t)=sin(n, IX/Lx)sin(ny Iy/Ly)cos(2T vt) Hence, the waves are quantized in two dimensions because their wavelengths must be constrained to cause A(x,y, t)to vanish at x =0 and x=Lx as well as at y =0 and y=Ly for all times t Let us now return to the issue of waves that describe electrons moving The pioneers of quantum mechanics examined functional forms similar to those shown above. For example, forms such as A= exp[+2 i(v t-x/)] were considered because they correspond to periodic waves that evolve in x and t under no external x-or t-dependent forces. Noticing that d2A/dx2=-(2m/)A and using the de broglie hypothesis 2=h/p in the above equation, one finds (2π/h)2A If a is supposed to relate to the motion of a particle of momentum p under no external forces(since the waveform corresponds to this case), p can be related to the energy e of the particle by E= p/2m. So, the equation for A can be rewritten as d2A/dx2=-2mE(2π/h)A
14 A(x,y,t) = sin(nx px/Lx ) sin(ny py/Ly ) cos(2p nt). Hence, the waves are quantized in two dimensions because their wavelengths must be constrained to cause A(x,y,t) to vanish at x = 0 and x = Lx as well as at y = 0 and y = Ly for all times t. Let us now return to the issue of waves that describe electrons moving. The pioneers of quantum mechanics examined functional forms similar to those shown above. For example, forms such as A = exp[±2pi(n t – x/l)] were considered because they correspond to periodic waves that evolve in x and t under no external x- or t- dependent forces. Noticing that d 2A/dx2 = - (2p/l) 2 A and using the de Broglie hypothesis l = h/p in the above equation, one finds d 2A/dx2 = - p2 (2p/h)2 A. If A is supposed to relate to the motion of a particle of momentum p under no external forces (since the waveform corresponds to this case), p2 can be related to the energy E of the particle by E = p2 /2m. So, the equation for A can be rewritten as: d 2A/dx2 = - 2m E (2p/h)2 A
or, alternatively -(h/2)2d2 turning to the time-dependence of A(x, t) and using v=E/h, one can also show that i(h/2π)dAdt=EA which, using the first result, suggests that i(h/2)dA/dt=-(h/2)d-A/dx a primitive form of the Schrodinger equation that we will address in much more detail below. Briefly, what is important to keep in mind that the use of the de broglie and Planck/Einstein connections(n-h/p and e=h v), both of which involve the constant h, produces suggestive connections between i(h/2π)d/ dt ande and between
15 or, alternatively, - (h/2p) 2 d2A/dx2 = E A. Returning to the time-dependence of A(x,t) and using n = E/h, one can also show that i (h/2p) dA/dt = E A, which, using the first result, suggests that i (h/2p) dA/dt = - (h/2p) 2 d2A/dx2 . This is a primitive form of the Schrödinger equation that we will address in much more detail below. Briefly, what is important to keep in mind that the use of the de Broglie and Planck/Einstein connections (l = h/p and E = h n), both of which involve the constant h, produces suggestive connections between i (h/2p) d/dt and E and between