Solved Problem on Schrödinger Equation

If the functions Ψ₁(x, t), Ψ₂(x, t) and Ψ₃(x, t) are solutions of the Schrödinger Equation for a potential V(x, t), show that the linear combination Ψ(x, t) = c₁Ψ₁(x, t) + c₂Ψ₂(x, t) + c₃Ψ₃(x, t) is also a solution of this equation.

Solution:

Applying the Schrödinger Equation

\[ \begin{gather} \bbox[#99CCFF,10px] {-{\frac{\hbar^2}{2m}}\frac{\partial^2\Psi(x,t)}{\partial x^2}+V(x,t)\Psi(x,t)=i\hbar\frac{\partial\Psi(x,t)}{\partial t}} \end{gather} \]

we want to show that the given function Ψ(x, t) is a solution of the Schrödinger Equation by expressing it in the following form

\[ \begin{gather} -{\frac{\hbar^2}{2m}}\frac{\partial^2\Psi(x,t)}{\partial x^2}+V(x,t)\Psi(x,t)-i\hbar\frac{\partial\Psi(x,t)}{\partial t}=0 \tag{I} \end{gather} \]

substituting the function Ψ(x, t) given in the problem into equation (I)

\[ \begin{align} & -{\frac{\hbar^2}{2m}}\frac{\partial^2[c_1\Psi_1(x,t)+c_2\Psi_2(x,t)+c_3\Psi_3(x,t)]}{\partial x^2}+\qquad\qquad\qquad\\ &\qquad +V(x,t)[c_1\Psi_1(x,t)+c_2\Psi_2(x,t)+c_3\Psi_3(x,t)]-\\ &\qquad -i\hbar\frac{\partial[c_1\Psi_1(x,t)+c_2\Psi_2(x,t)+c_3\Psi_3(x,t)]}{\partial t}=0 \\[10pt] & -{\frac{\hbar^2}{2m}}\left[c_1\frac{\Psi_1(x,t)}{\partial x^2}+c_2\frac{\Psi_2(x,t)}{\partial x^2}+c_3\frac{\Psi_3(x,t)}{\partial x^2}\right]+\qquad\quad\qquad\\ &\qquad +c_1\Psi_1(x,t)V(x,t)+c_2\Psi_2(x,t)V(x,t)+c_3\Psi_3(x,t)V(x,t)+\\ &\qquad -c_1 i\hbar\frac{\partial \Psi_1(x,t)}{\partial t}-c_2 i\hbar\frac{\partial\Psi_2(x,t)}{\partial t}-c_3 i\hbar\frac{\partial \Psi_3(x,t)}{\partial t}=0 \\[10pt] & c_1\;\left[-{\frac{\hbar^2}{2m}}\frac{\partial^2\Psi_1(x,t)}{\partial x^2}+V(x,t)\Psi_1(x,t)-i\hbar\frac{\partial \Psi_1(x,t)}{\partial t}\right]+\qquad\quad\\ &\qquad +c_2\;\left[-{\frac{\hbar^2}{2m}}\frac{\partial^2\Psi_2(x,t)}{\partial x^2}+V(x,t)\Psi_2(x,t)-i\hbar\frac{\partial \Psi_2(x,t)}{\partial t}\right]+\quad \tag{II} \\ &\qquad +c_3\;\left[-{\frac{\hbar^2}{2m}}\frac{\partial^2\Psi_3(x,t)}{\partial x^2}+V(x,t)\Psi_3(x,t)-i\hbar\frac{\partial \Psi_3(x,t)}{\partial t}\right]=0 \end{align} \]

Since the problem states that the functions Ψ₁(x, t), Ψ₂(x, t) and Ψ₃(x, t) are solutions of the Schrödinger Equation, this means that

\[ \begin{gather} -{\frac{\hbar^2}{2m}}\frac{\partial^2\Psi_1(x,t)}{\partial x^2}+V(x,t)\Psi_1(x,t)-i\hbar\frac{\partial\Psi_1(x,t)}{\partial t}=0 \\[10pt] -{\frac{\hbar^2}{2m}}\frac{\partial^2\Psi_2(x,t)}{\partial x^2}+V(x,t)\Psi_2(x,t)-i\hbar\frac{\partial \Psi_2(x,t)}{\partial t}=0 \\[10pt] -{\frac{\hbar^2}{2m}}\frac{\partial^2\Psi_3(x,t)}{\partial x^2}+V(x,t)\Psi_3(x,t)-i\hbar\frac{\partial\Psi_3(x,t)}{\partial t}=0 \end{gather} \]

substituting these values into equation (II)

\[ \begin{gather} c_1\times 0+c_2\times 0+c_3\times 0=0 \end{gather} \]

this equality will hold for any values of c₁, c₂ and c₃, making equation (I) true. Therefore, Ψ(x, t) is a solution of the Schrödinger Equation .

Fisicaexe - Physics Solved Problems by Elcio Brandani Mondadori is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License .