Ch. 8 Schrödinger Equation PDF

**Ch. 8** **Schrödinger Equation** In section **4.1**, we had a brief review about early stages of quantum mechanics. In this chapter, after a mathematics summary of wave, we deal with the equation of particle wave; so-called Schrödinger Equation. And then we will discuss a little bit about its sol...

**Ch. 8** **Schrödinger Equation** In section **4.1**, we had a brief review about early stages of quantum mechanics. In this chapter, after a mathematics summary of wave, we deal with the equation of particle wave; so-called Schrödinger Equation. And then we will discuss a little bit about its solution for hydrogen atom. **8.1 Wave Function and Wave Equation** A graph of sine function; (part of [*y* = sin *x*]{.math.inline}, [\$\\sin\\left( x - \\frac{\\pi}{5} \\right)\$]{.math.inline}, [\$\\sin\\left( x - \\frac{2\\pi}{5} \\right)\$]{.math.inline}, [\$\\sin\\left( x - \\frac{6\\pi}{5} \\right)\\ \$]{.math.inline}) The **wavelength** of waveform [*y* = *sinx* ]{.math.inline} or [*y* = *A* *sinx*]{.math.inline} is '[2*π*]{.math.inline}' ([*π* = 3.14159 .  .  )]{.math.inline} for a waveform, of which wavelength is []{.math.inline} ; \ [\$\${y = sin}\\frac{2\\pi}{}x = \\sin\\text{kx}\$\$]{.math.display}\ where **wavenumber** is defined as [\$k = \\frac{2\\pi}{}\$]{.math.inline} Now, the **frequency** *f* is the number of oscillations in a second; \ [\$\$f = \\frac{1}{T}\$\$]{.math.display}\ where *T* is the **period** of oscillation. Angular frequency is defined as; [*ω* = 2*πf*]{.math.inline}. Then, the **speed** (**\|velocity\|**) of wave is given as; \ [*v* = *f* ]{.math.display}\ Let us exemplify the wave properties. Given the speed of sound at room temperature as [340 *m*/*s*]{.math.inline}, then what is the wavelength of 440 Hz sound? [*v* = *f*]{.math.inline} =\> [340 = 440 ]{.math.inline} Then we find the wavelength [ = 0.773 *m*]{.math.inline}. Audible frequency range is from 20 to 20000 Hz, of which corresponding wavelengths are 17 m to 1.7 cm. Speed of light (EM wave) is [3 × 10^8^]{.math.inline} m/s. FM radio signal is carried by about 100 MHz, of which wavelength is 3 meters. Cell phones use frequency band between 450 and 2700 MHz. The corresponding wavelength range is 11 - 67 cm. Human eyes can recognize light of wavelength 400 to 700 nm, of which frequency range is [750 ∼ 430]{.math.inline} THz. Consider the wave function for a waveform [*A*sin kx]{.math.inline} travels with velocity *v* in the direction of *x*-axis. [*y*(*x*) = *A*sin kx]{.math.inline} =\> [*y*(*x*, *t*) = *A*sin *k*(*x*−*vt*) = *A*sin (*kx* − *ωt*)]{.math.inline} while the wave traveling in the opposite direction can be represented by a wave function; \ [*y*(*x*, *t*) = *A*sin *k*(*x*+*vt*) = *A*sin (*kx* + *ωt*)]{.math.display}\ Likewise we have another traveling wave function as; \ [*y*(*x*, *t*) = *A*cos *k*(*x*∓*vt*) = *A*cos (*kx* ∓ *ωt*)]{.math.display}\ Using Euler Formula [*e*^iθ^ = cos *θ* + *i*sin *θ*]{.math.inline} (supplementary), often wave function is expressed as; \ [*ψ*(*x*,*t*) = *A* *e*^*i*(*kx* − *ωt*)^]{.math.display}\ **One-Dimensional Wave Equation** Many different kinds of waves exist, for example, EM wave, mechanical wave, sound wave, etc. And the 1-dimensional wave function satisfies the following equation; \ [\$\$\\frac{\\partial\^{2}\\psi}{\\partial x\^{2}} = \\ \\frac{1}{v\^{2}}\\ \\frac{\\partial\^{2}\\psi}{\\ \\partial t\^{2}}\\ \$\$]{.math.display}\ **Two-Dimensional Wave Function and Wave Equation** 2-dimensional wave function can be written as; \ [*ψ*(*x*,*y*,*t*) = *A* *e*^*i*(*k*~*x*~ *x*+ *k*~*y*~ *y*  − *ωt*)^]{.math.display}\ And 2-dimensional wave function satisfies the following equation; \ [\$\$\\frac{\\partial\^{2}\\psi}{\\partial x\^{2}} + \\frac{\\partial\^{2}\\psi}{\\partial y\^{2}} = \\ \\frac{1}{v\^{2}}\\ \\frac{\\partial\^{2}\\psi}{\\ \\partial t\^{2}}\\ \$\$]{.math.display}\ where wave vector [*k⃗*= *k*~*x*~ *x̂* + *k*~*y*~ *ŷ*]{.math.inline} with its magnitude [\$k = \\sqrt{k\_{x}\^{2} + k\_{y}\^{2}}\\ \$]{.math.inline}, and [\$v = \\frac{\\omega}{k}\$]{.math.inline}. **Three-Dimensional Wave Function and Wave Equation** 3-dimensional wave function can be written as; \ [*ψ*(*x*,*y*,*z*,*t*) = *A* *e*^*i*(*k*~*x*~\ x+ *k*~*y*~ *y* + *k*~*z*~ *z*  − *ωt*)^]{.math.display}\ And 3-dimensional wave function satisfies the following equation; \ [\$\$\\frac{\\partial\^{2}\\psi}{\\partial x\^{2}} + \\frac{\\partial\^{2}\\psi}{\\partial y\^{2}} + \\frac{\\partial\^{2}\\psi}{\\partial z\^{2}} = \\ \\frac{1}{v\^{2}}\\ \\frac{\\partial\^{2}\\psi}{\\ \\partial t\^{2}}\\ \$\$]{.math.display}\ where wave vector [*k⃗*= *k*~*x*~ *x̂* + *k*~*y*~ *ŷ* + *k*~*z*~ *ẑ*]{.math.inline} with its magnitude [\$k = \\sqrt{k\_{x}\^{2} + k\_{y}\^{2} + k\_{z}\^{2}}\\ \$]{.math.inline}, and [\$v = \\frac{\\omega}{k}\$]{.math.inline}. The wave vector [*k⃗*]{.math.inline} in two or three-dimension also indicates the direction of wave propagation. \ Euler formula; [*e*^iθ^ = cos *θ* + *i*sin *θ*]{.math.inline} \ [\$\$\\sin x = x - \\ \\frac{x\^{3}}{3!} + \\ \\frac{x\^{5}}{5!} - \\frac{x\^{7}}{7!} + \\.\\ \\.\\ \\.\$\$]{.math.display}\ \ [\$\$\\cos x = 1 - \\ \\frac{x\^{2}}{2!} + \\ \\frac{x\^{4}}{4!} - \\frac{x\^{6}}{6!} + \\.\\ \\.\\ \\.\$\$]{.math.display}\ \ [\$\$e\^{x} = 1 + x + \\ \\frac{x\^{2}}{2!} + \\ \\frac{x\^{3}}{3!} + \\frac{x\^{4}}{4!} + \\.\\ \\.\\ \\.\$\$]{.math.display}\ Then we have \ [\$\$e\^{\\text{iθ}} = 1 + i\\theta + \\ \\frac{{(i\\theta)}\^{2}}{2!} + \\ \\frac{{(i\\theta)}\^{3}}{3!} + \\frac{{(i\\theta)}\^{4}}{4!} + \\frac{{(i\\theta)}\^{5}}{5!} + \\ \\frac{{(i\\theta)}\^{6}}{6!}\\text{\\.\\ \\.\\ \\.}\$\$]{.math.display}\ \ [\$\$\\ = 1 - \\frac{\\ \\theta\^{2}}{2!} + \\frac{\\text{\\ θ}\^{4}}{4!} - \\.\\ \\.\\ \\.\\ \\ \\ + \\ \\ i\\ \\lbrack\\ \\theta - \\ \\frac{\\text{\\ θ}\^{3}}{3!} + \\ \\frac{\\theta\^{5}}{5!} - \\ \\.\\ \\.\\ \\.\\ \\rbrack\$\$]{.math.display}\ [ = *cosθ* + *i*sin *θ*]{.math.inline} ( [*i*^2^ =  − 1 ,  *i*^3^ =  − *i* ,  *i*^4^ = 1 ,  *i*^5^ = *i* ,  *i*^6^ =  − 1 ,   .  .  .  )]{.math.inline} **8.2** **Schrödinger Equation** As summarized before, the underlying concepts of quantum mechanics are; Planck's constant, energy quanta, and de Broglie wave. \ [*h* = 6.626  × 10^ − 34^ \[J s\]]{.math.display}\ \ [*E* = *hf*]{.math.display}\ \ [\$\$ = \\frac{h}{p} = \\frac{h}{\\text{mv}}\$\$]{.math.display}\ *Matter was neither particle nor wave, but a bit of both - a wavicle also called wave packet. Attention shifted from providing images for wavicles to sorting out how they behaved. The quest quickly attained its goal, and the central equation of quantum theory emerged. The equation bears the name of Erwin* *Schrödinger. - Ian Stewart* Here we state the particle-wave equation of Schrödinger first, and then interpret its meaning by substitution of wave function into the equation. Schrödinger equation in three dimension is given as follows; \ [\$\$- \\frac{h\^{2}}{8\\pi\^{2}\\text{\\ m}}\\left\\lbrack \\ \\frac{\\partial\^{2}\\psi}{\\partial x\^{2}} + \\frac{\\partial\^{2}\\psi}{\\partial y\^{2}} + \\frac{\\partial\^{2}\\psi}{\\partial z\^{2}} \\right\\rbrack + V\\ \\psi = \\ \\ i\\ \\frac{h}{2\\pi}\\ \\frac{\\partial\\psi}{\\partial t}\$\$]{.math.display}\ Write the wave function as a plane wave; \ [*ψ*(*x*,*y*,*z*,*t*) = *A* *e*^*i*(*k*~*x*~ *x*+ *k*~*y*~ *y* + *k*~*z*~ *z*  − *ωt*)^]{.math.display}\ If confined to one dimension only, then we have simplified version of the equation and solution as; \ [\$\$- \\frac{h\^{2}}{8\\pi\^{2}\\text{\\ m}}\\frac{\\partial\^{2}\\psi}{\\partial x\^{2}} + V\\ \\psi = \\ \\ \\text{i\\ }\\frac{h}{2\\pi}\\ \\frac{\\partial\\psi}{\\partial t}\$\$]{.math.display}\ \ [*ψ*(*x*,*t*) = *A* *e*^*i*(*kx*  − *ωt*)^]{.math.display}\ First, check; \ [\$\$\\frac{\\partial\^{2}\\psi}{\\partial x\^{2}} = \\ - k\^{2}\\ \\psi\$\$]{.math.display}\ Then we have \ [\$\$- \\frac{h\^{2}}{8\\pi\^{2}\\text{\\ m}}\\frac{\\partial\^{2}\\psi}{\\partial x\^{2}} = \\frac{h\^{2}}{8\\pi\^{2}\\text{\\ m}}\\ k\^{2}\\psi = \\ \\frac{1}{2}\\text{\\ m}v\^{2}\\psi\\ \$\$]{.math.display}\ \ [\$\$(\\ \\text{where}\\text{\\ \\ \\ \\ }\\frac{h}{\\text{mv}} = \\  = \\ \\frac{2\\pi}{k}\\text{\\ \\ \\ \\ \\ or\\ \\ \\ }\\frac{\\text{hk}}{2\\pi} = mv\\ \\ )\\ \\ \\ \$\$]{.math.display}\ Therefore, the left hand side of the Schrödinger equation becomes; \ [\$\$(\\frac{1}{2}\\text{\\ m}v\^{2} + V)\\psi\\ \$\$]{.math.display}\ Now, check; \ [\$\$\\frac{\\partial\\psi}{\\partial t} = \\ - i\\text{ω\\ ψ}\$\$]{.math.display}\ Then the right hand side; \ [\$\$\\text{i\\ }\\frac{h}{2\\pi}\\ \\frac{\\partial\\psi}{\\partial t} = \\frac{h}{2\\pi}\\text{\\ ω\\ }\\psi = hf\\ \\psi = E\\ \\psi\\ \$\$]{.math.display}\ Therefore, we see that; \ [\$\$(\\frac{1}{2}\\text{\\ m}v\^{2} + V)\\psi = E\\ \\psi\\ \$\$]{.math.display}\ which simply stands for; \ [\$\$\\frac{1}{2}\\text{\\ m}v\^{2} + V = E\$\$]{.math.display}\ 'kinetic energy + potential energy = total energy' **8.3 Semi-Classical Solution for Atomic Electron State** The solution for Schrödinger equation in three dimension is often acquired in spherical coordinates. However, it is rather elaborate, and so here, we circumvent the rigorous procedure and resort to semiclassical approach (still some main results are the same). de Broglie's wavelength; \ [\$\$ = \\frac{h}{\\text{mv}}\$\$]{.math.display}\ We claim the orbit of electron to be multiple of []{.math.inline} *;* \ [\$\$2\\pi r = n = \\frac{\\text{nh}}{\\text{mv}}\$\$]{.math.display}\ where [*n* = 1, 2, 3,   .  .  . ]{.math.inline} And the electrostatic attraction between the nucleus and the electron is acting as the centripetal force; \ [\$\$k\_{e}\\frac{e\\ (Ze)}{r\^{2}} = \\ \\frac{mv\^{2}}{r}\$\$]{.math.display}\ where *Z* is the atomic number (*Z*=1 for hydrogen) From these two conditions, we find \ [\$\$v = \\frac{2\\pi\\ k\_{e}\\text{\\ Z}e\^{2}}{\\text{nh}}\\text{\\ \\ \\ \\ \\ \\ \\ }\\text{and}\\ \\ \\ \\ \\ \\ \\ \\ r = \\frac{n\^{2}\\ h\^{2}}{4\\pi\^{2}\\text{\\ m\\ }k\_{e}\\text{\\ Z\\ }e\^{2}} = \\frac{n\^{2}}{Z}a\_{B}\$\$]{.math.display}\ ([*a*~*B*~=]{.math.inline} Bohr radius) The electric potential energy is; \ [\$\$V = \\ - k\_{e}\\frac{\\text{\\ Z}e\^{2}}{r}\$\$]{.math.display}\ We have the total energy [\$E = \\frac{1}{2}\\text{\\ m}v\^{2} + V\\ \$]{.math.inline} ; \ [\$\$E = \\ - \\frac{2\\pi\^{2}\\text{m\\ }k\_{e}\^{2}\\ Z\^{2}e\^{4}}{h\^{2}}\\ \\times \\frac{1}{n\^{2}} = \\ - \\ Z\^{2}\\ E\_{R} \\times \\frac{1}{n\^{2}}\$\$]{.math.display}\ where is Rydberg energy [*E*~*R*~ = 2.18 × 10^ − 18^*J*]{.math.inline}. (also [*E*~*R*~ = 13.6 *eV*]{.math.inline} where [1 *eV* = 1.6 × 10^ − 19^ J]{.math.inline}) (exactly same energy levels are found by solving the full 3-dim Schrödinger equation in spherical coordinates!) For the electron bound to hydrogen atom, its energy levels in \[eV\] are determined as; \ [\$\$- 13.6,\\ - 13.6 \\times \\frac{1}{4} = - 3.4,\\ - 13.6 \\times \\frac{1}{9} = - 1.51,\\ \\ \\.\\ \\.\\ \\.\\ \$\$]{.math.display}\ that is [*E*~1~ =  − 13.6,  E~2~ =  − 3.4,  *E*~3~ =  − 1.51,  *E*~4~ =  − 0.85]{.math.inline} \[eV\] Energy levels of ground state, 1st excited state, 2nd excited state,... Following figure describes these energy levels and transition between different energy levels.  It has been found that a photon is emitted when the electron state is changed from high energy state (E~high~) to low energy state (E~low~). And the frequency *f* of the photon is given as; \ [**hf=** *Δ***E=** **E**~high~**−E**~low~]{.math.display}\ As an example, if electron state of [*E*~3~ =  − 1.51 eV]{.math.inline} is changed to[ *E*~2~ =  − 3.40 eV]{.math.inline}, then the frequency of emitted photon should be; (Planck constant [*h* = 6.626 × 10^ − 34^ \[Js\]]{.math.inline}) \ [*hf*= *ΔE* = (3.40−1.51) = 1.89 eV = 1.89 × 1.6 × 10^ − 19^*J*]{.math.display}\ \ [*f* = 4.56 × 10^15^]{.math.display}\ Corresponding wavelength is; \ [\$\$ = \\frac{c}{f} = \\frac{3 \\times 10\^{8}}{4.56 \\times 10\^{15}} = 6.25 \\times 10\^{- 7}\\ \\left\\lbrack m \\right\\rbrack = 625\\ \\lbrack\\text{nm}\\rbrack\$\$]{.math.display}\ which corresponds to visible light of red color. **8.4 Solution of the full 3-dim Schrödinger equation in Spherical Coordinates** Schrödinger equation is a [partial differential equation](https://en.wikipedia.org/wiki/Partial_differential_equation) for the [wave behavior](https://en.wikipedia.org/wiki/Wave_function) of a quantum-mechanical system. The equation is named after [Erwin Schrödinger](https://en.wikipedia.org/wiki/Erwin_Schr%C3%B6dinger), who postulated it in 1925. Time-dependent Schrödinger equation (shown in the former section **8.2**) is the equation of motion, while time-independent Schrödinger equation (shown below) is for stationary wave function of quantized energy. Assuming time dependence as [*e*^ − *iωt*^]{.math.inline}, the right hand side of the equation becomes; [\$\\text{i\\ }\\frac{h}{2\\pi}\\ \\frac{\\partial\\psi}{\\partial t} = \\frac{h}{2\\pi}\\ \\omega\\ \\psi = hf\\ \\psi = E\\ \\psi\$]{.math.inline}, then we have; \ [\$\$- \\frac{h\^{2}}{8\\pi\^{2}\\text{\\ m}}\\left\\lbrack \\ \\frac{\\partial\^{2}\\psi}{\\partial x\^{2}} + \\frac{\\partial\^{2}\\psi}{\\partial y\^{2}} + \\frac{\\partial\^{2}\\psi}{\\partial z\^{2}} \\right\\rbrack + V\\ \\psi = \\ E\\ \\psi\$\$]{.math.display}\ To know the state of electron bound to atom, it is suitable to use spherical coordinate, and then the Schrödinger equation becomes; \ [\$\$- \\frac{h\^{2}}{8\\pi\^{2}\\text{\\ m}}\\left\\lbrack \\frac{1}{r}\\frac{\\partial\^{2}(r\\psi)}{\\partial r\^{2}} + \\frac{1}{r\^{2}\\sin\\theta}\\frac{\\partial}{\\partial\\theta}(\\sin{\\text{θ\\ }\\frac{\\partial\\psi}{\\partial\\theta})} + \\frac{1}{{(r\\sin\\theta)}\^{2}}\\frac{\\partial\^{2}\\psi}{\\partial\\varphi\^{2}} \\right\\rbrack + V\\ \\psi = \\ E\\ \\psi\$\$]{.math.display}\ with the potential [\$V = \\ - \\frac{k\_{e}Ze\^{2}}{r}\$]{.math.inline}. This equation can be solved by separation of variables by writing [*ψ*(*r*, *θ*, *φ*) = *R*(*r*)(*θ*)(*φ*)]{.math.inline}, however, we hereby only describe the result - characteristics of the solution. Write [(*φ*) = *e*^imφ^]{.math.inline}*, then we should have m* to be integer for physical reality. And we find another quantization from the equation for [(*θ*)]{.math.inline} such as; \ [\$\$\\frac{1}{\\sin\\theta}\\frac{d}{\\text{dθ}}(\\sin{\\text{θ\\ }\\frac{d}{\\text{dθ}}) + \\left( l\\left( l + 1 \\right) - \\frac{m\^{2}}{\\left( \\sin\\theta \\right)\^{2}} \\right) = 0}\$\$]{.math.display}\ where *l* can be any positive integer; 0, 1, 2, 3... The remaining equation for *r* is given as; \ [\$\$\\frac{1}{r}\\frac{d\^{2}}{\\text{dr}\^{2}}\\left( \\text{rR} \\right) = \\frac{8\\pi\^{2}\\text{\\ m}}{h\^{2}}\\left\\lbrack \\text{\\ V}\\left( r \\right) - E \\right\\rbrack R\\left( r \\right) + \\frac{l(l + 1)}{r\^{2}}R(r)\$\$]{.math.display}\ Then from the equation for *r*, we find the energy quantization; \ [\$\$E = \\ - \\frac{2\\pi\^{2}\\text{m\\ }k\_{e}\^{2}\\ Z\^{2}e\^{4}}{h\^{2}}\\ \\times \\frac{1}{n\^{2}} = \\ - \\ Z\^{2}\\ E\_{R} \\times \\frac{1}{n\^{2}}\$\$]{.math.display}\ which is the same formula already found by semi-classical approach. The **probability density** of electron is given as [**P**(**r**)**=\|ψ\|**^**2**^.]{.math.inline} For ground state, *n*=1, the possible value of *l* is *l*=0 and then we also have *m*=0; 1s state. For the first excited level, *n*=2, possible set of values of *l* and *m* are (0,0), (1,1), (1,0), (1,-1); 2s state and three 2p states. For the second excited level, *n*=3, possible set of values of *l* and *m* are (0,0), (1,1), (1,0), (1,-1), (2,2), (2,1), (2,0), (2,-1), (2,-2); 3s state, three 3p states and five 3d states. The letters s, p, d, f stand for *l*=0, 1, 2, 3 respectively. Total number of states for energy quantum number *n* is found as; 1 (*n*=1), 4 (*n*=2), 9 (*n*=3), 16 (*n*=4)... With doubling the number of states (two electrons occupy each same set of quantum numbers) we have; 2 (*n*=1), 8 (*n*=2), 18 (*n*=3), 32 (*n*=4)... Following three figures describes [*P*(*r*) = \|*ψ*\|^2^]{.math.inline} for hydrogen.  *As Feynman explained for light rays, the classical world emerges when you superpose all of the quantum alternatives. There is only one such superposition, so there is only one classical universe. Ours. -- Ian Stewart* *Quantum mechanics isn't confined to the laboratory. The whole of modern electronics depends on it. -- Ian Stewart* (supplementary) with two notations \ [\$\$ = \\frac{h}{2\\pi}\$\$]{.math.display}\ \ [\$\$\\nabla\^{2}\\psi = \\frac{\\partial\^{2}\\psi}{\\partial x\^{2}} + \\frac{\\partial\^{2}\\psi}{\\partial y\^{2}} + \\frac{\\partial\^{2}\\psi}{\\partial z\^{2}}\$\$]{.math.display}\ Schrödinger Equation can be written as \ [\$\$- \\frac{\^{2}}{2m}\\nabla\^{2}\\psi + V\\ \\psi = \\ i\\frac{\\partial\\psi}{\\partial t}\$\$]{.math.display}\ \-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-- **References** Modern Physics for Scientists and Engineers, Taylor & Zafiratus, Prentice-Hall In Pursuit of the Unknown -17 Equations that Changed the World, Ian Stewart, Basic Books NY [[https://en.wikipedia.org/wiki/Schrödinger\_equation]](https://en.wikipedia.org/wiki/Thermodynamics) \* The content of this chapter is abridged from the books and other sources as well as Wikipedia. **\[Homework\]** \[1\] Find the frequency and wavelength of photon emitted by transition from [*E*~2~]{.math.inline} to [*E*~1~]{.math.inline} of electron state bound to hydrogen atom. (use SI units)