# Linearity of Schrodinger Equation

• dainylee
In summary, the conversation revolves around demonstrating the superposition principle of the Schrodinger equation algebraically. The main question is how to show that if Psi1(x,t) and Psi2(x,t) are both solutions, then Psi(x,t) = Psi1(x,t) + Psi2(x,t) is also a solution. The conversation also discusses using coefficients a and b to extend the result. The desired outcome is to show that -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} \Psi(x,t) + V(x) \Psi(x,t) = i \hbar \frac{\partial}{\partial t} \Psi(x,t) is true by substit
dainylee
How do you algebraically demonstrate the superposition principle revealed by the Schrodinger equation (ie. If Psi1(x,t) and Psi2(x,t) are both solutions then Psi(x,t)= Psi1(x,t)+Psi2(x,t) is also a solution.)?

What do you get when you substitute Psi(x,t) into the left side of the Schrodinger equation?

I am not quite sure what you are asking me to do...Psi(x,t) is already in there?

No, $$\psi(x,t)$$ is not in there. However, you do know that $$\psi_1(x,t)$$ is a solution -

$$-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}\psi_1(x,t)+V(x)\psi_1(x,t)=i \hbar \frac{d}{dt}\psi_1(x,t)$$And $$\psi_2(x,t)$$ is also in there:

$$-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}\psi_2(x,t)+V(x)\psi_2(x,t)=i \hbar \frac{d}{dt}\psi_2(x,t)$$However, you have to show that $$\psi=\psi_1+\psi_2$$ also satisfies this dynamic equation.

And then you can extend your result to $$a\psi_1 + b\psi_2$$ for any coefficents a,b.

Last edited by a moderator:
dainylee said:
I am not quite sure what you are asking me to do...Psi(x,t) is already in there?

You have to show that

$$-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} \Psi(x,t) + V(x) \Psi(x,t) = i \hbar \frac{\partial}{\partial t} \Psi (x,t).$$

In the left, substitute $\Psi = \Psi_1 + \Psi_2$, and, using that $\Psi_1$ and $\Psi_2$ both statisfy Schrodinger's equation, work your way to the right side of the equation you must show true.

## What is the Schrodinger equation?

The Schrodinger equation is a fundamental equation in quantum mechanics that describes how the quantum state of a physical system changes over time. It was developed by Austrian physicist Erwin Schrodinger in 1925.

## What is linearity in the context of the Schrodinger equation?

In the context of the Schrodinger equation, linearity refers to the property that the equation follows the principle of superposition. This means that the overall solution to the equation is equal to the sum of individual solutions, and the equation is linear in terms of its dependent variable.

## Why is linearity important in the Schrodinger equation?

Linearity is important in the Schrodinger equation because it allows for the combination of multiple solutions to describe a more complex physical system. This allows for a more accurate description of quantum systems and makes the equation a powerful tool in understanding the behavior of particles on a microscopic level.

## What are the implications of linearity in the Schrodinger equation?

The implications of linearity in the Schrodinger equation are far-reaching. It allows for the prediction of the behavior of quantum systems and is the basis for many important concepts in quantum mechanics, such as superposition and entanglement. It also allows for the development of advanced technologies, such as quantum computers and quantum cryptography.

## How is the linearity of the Schrodinger equation tested and verified?

The linearity of the Schrodinger equation has been extensively tested and verified through experiments and observations. These include the double-slit experiment, which demonstrates the principle of superposition, and quantum entanglement experiments, which confirm the linearity of the equation in terms of entangled particles. The equation has also been successfully used to make accurate predictions about the behavior of quantum systems in real-world scenarios.

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