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Clever-Name said:You're almost there. What do you know about a linear combination of solutions to a differential equation? What does that say about each independent solution?
ideasrule said:A differential equation is linear if none of the terms are being squared, cubed, square rooted, etc. Schrodinger's equation is linear because the wavefunction isn't being raised to any power, and neither is its second derivative.
For a linear system of equations (not necessarily differential equations), if you know that A and B are both solutions, all linear combinations of A and B must also be solutions. Here, you need to prove that if ψ is a solution to Schrodinger's equation, so is ψ*. Then you'll know that ψ+ψ* and i(ψ−ψ∗) are also solutions to Schrodinger's equation. But these are both real, which completes the proof.
vela said:Let [itex]\psi(x) = u(x) + i v(x)[/itex] where u(x) and v(x) are real functions. Then [itex]\psi^*(x) = u(x) - i v(x)[/itex].
What are [itex]\psi(x)+\psi^*(x)[/itex] and [itex]i(\psi(x)-\psi^*(x))[/itex] equal to?
Wait a minute, it is easy to solve because of the hint.vela said:Let [itex]\psi(x) = u(x) + i v(x)[/itex] where u(x) and v(x) are real functions. Then [itex]\psi^*(x) = u(x) - i v(x)[/itex].
What are [itex]\psi(x)+\psi^*(x)[/itex] and [itex]i(\psi(x)-\psi^*(x))[/itex] equal to?
A wave function is a mathematical function that describes the probability of a particle being in a certain location in space at a given time.
Wave functions are used in quantum mechanics to describe the behavior of subatomic particles, such as electrons. They help scientists predict the location of particles and their properties.
In science, proving a wave function means that experimental data and mathematical calculations support the predicted behavior of a particle according to the wave function.
Being stumped means that the experimental data and mathematical calculations do not align with the predicted behavior of a particle according to the wave function. This can indicate a need for further research or a potential flaw in the current understanding of the particle's behavior.
To overcome being stumped, a scientist may need to revise the wave function or conduct further experiments to gather more data. They may also need to collaborate with other scientists and consider alternative explanations for the discrepancies. Ultimately, it may require a re-evaluation of scientific theories and understanding in order to fully explain the behavior of the particle.