Question

If the function psi1 & psi2 are solution of Schrödinger wave equation for a particle , then prove that a1psi1+a2psi2 is also a solution of same equation, where a1 & a2 are arbitrary constants.

Answer 1

Answer on Question #45544, Physics, Quantum Mechanics

If the function $\psi_{1}$ and $\psi_{2}$ are solution of Schrodinger wave equation for a particle , then prove that $a_{1}\psi_{1}+a_{2}\psi_{2}$ is also a solution of same equation, where $a_{1}$ and $a_{2}$ are arbitrary constants.

Solution

see on next page.

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Linearity in $\Psi(x,t)$ : A linear combination $\Psi(x,t)$ of two solutions $\Psi_1(x,t)$ and $\Psi_2(x,t)$ is also a solution. $\Psi(x,t) = c_1\Psi_1(x,t) + c_2\Psi_2(x,t)$ E

$\Psi_{1}(x,t)$ is a solution and thus satisfies: $\mathbf{E}_1$ $\frac{\hbar^2}{2m}\frac{\partial^2\Psi_1(x,t)}{\partial x^2} +V(x,t)\Psi_1(x,t) = i\hbar \frac{\partial\Psi_1(x,t)}{\partial t}$

$\Psi_{2}(x,t)$ is a solution and thus satisfies: $\mathbf{E}_2$ $-\frac{\hbar^2}{2m}\frac{\partial^2\Psi_2(x,t)}{\partial x^2} +V(x,t)\Psi_2(x,t) = i\hbar \frac{\partial\Psi_2(x,t)}{\partial t}$

Add Eqs. $\mathbf{E}_1$ and $\mathbf{E}_2$ together as $c_{1}\mathbf{E}_{1} + c_{2}\mathbf{E}_{2}$

$c_{1}\left[-\frac{\hbar^{2}}{2m}\frac{\partial^{2}\Psi_{1}(x,t)}{\partial x^{2}} +V(x,t)\Psi_{1}(x,t)\right] + c_{2}\left[-\frac{\hbar^{2}}{2m}\frac{\partial^{2}\Psi_{2}(x,t)}{\partial x^{2}} +V(x,t)\Psi_{2}(x,t)\right] = c_{1}\left[i\hbar \frac{\partial\Psi_{1}(x,t)}{\partial t}\right] + c_{2}\left[i\hbar \frac{\partial\Psi_{2}(x,t)}{\partial t}\right]$

Rearrange a bit:

$-\frac{\hbar^2}{2m}\left[c_1\frac{\partial^2\Psi_1(x,t)}{\partial x^2} +c_2\frac{\partial^2\Psi_2(x,t)}{\partial x^2}\right] + V(x,t)\left[c_1\Psi_1(x,t) + c_2\Psi_2(x,t)\right] = i\hbar \left[c_1\frac{\partial\Psi_1(x,t)}{\partial t} +c_2\frac{\partial\Psi_2(x,t)}{\partial t}\right]$

Differentiation is linear:

$-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} [c_1\Psi_1(x,t) + c_2\Psi_2(x,t)] + V(x,t)[c_1\Psi_1(x,t) + c_2\Psi_2(x,t)] = i\hbar \frac{\partial}{\partial t} [c_1\Psi_1(x,t) + c_2\Psi_2(x,t)]$

Substitute Eqn. $\mathbf{E}_3$ to recover the Schrödinger equation for $\Psi(x,t)$ thus showing that $\Psi(x,t)$ is also a solution. $-\frac{\hbar^2}{2m}\frac{\partial^2\Psi(x,t)}{\partial x^2} + V(x,t)\Psi(x,t) = i\hbar \frac{\partial\Psi(x,t)}{\partial t}$

Figure 1: a