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Time dependent Schrodinger equation

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)
$$\phi(t) = C_+ exp(-i2 \pi \frac{E}{\hbar 2 \pi} t)  + C_- exp(i2 \pi \frac{E}{\hbar 2 \pi} t)$$
$$\phi(t) = C_+ exp(-i\frac{E}{\hbar}t) + C_- exp(i \frac{E}{\hbar} t)$$ ..........(9)
For the present special case,
$$C_- = 0$$ ........(10)
Substitute equ (10) into equ (9), we have:
$$phi(t) = C_+ exp(-i \frac {E}{\hbar} t)$$ .......(11)
Substitute equ (11) into equ (3), we have:
$$\Psi(r,t) = C_+ \psi(r) exp(-i\frac{E}{\hbar}t)$$ ...........(12)
Differentiating equ (12) with respect to t and rearranging, we have:
$$\Psi(r,t) = i \frac{\hbar}{E} \frac{d \Psi (r,t)}{dt}$$ ..............(13)
Substitute equ (13) into the right hand side of equ (4), we have:
$$-\frac{\hbar^{2}}{2m} \nabla^{2} \Psi (r,t) + V(r) \Psi (r,t) = i\hbar \frac{d \Psi (r,t)}{dt}$$ ............(14)
Equ (14) is the time dependent Schrodinger equation.

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