 Physics Help Forum Show that psi(-x) is a solution with the same eigenvalue as psi(x)
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 Quantum Physics Quantum Physics Help Forum Sep 26th 2018, 07:03 AM #1 Junior Member   Join Date: Sep 2018 Posts: 2 Show that psi(-x) is a solution with the same eigenvalue as psi(x) I have been given the assignment to show that ψ(-x) is another solution with the same energy eigenvalue as ψ(x), given that the potential is even around x=0. The solutions are one-dimensional. I have tried to get my head around it but I am not sure what to look for. My idea would be to divide through by the wave function in the wave equation, and we would then be left with the energy-eigenvalue E as a function of the potential, which we know is even around the origin: -ℏ^2/2m (d^2 ψ(x))/(dx^2 )+V(x)ψ(x)=ψ(x)E. Dividing by ψ(x): -ℏ^2/2m (d^2)/(dx^2 )+V(x)=E And the eigenvalue is now a function of the potential only. Is this an illegal move? Or where do I go from here?   Sep 26th 2018, 12:46 PM   #2
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 I have been given the assignment to show that ψ(-x) is another solution with the same energy eigenvalue as ψ(x), given that the potential is even around x=0. The solutions are one-dimensional. I have tried to get my head around it but I am not sure what to look for. My idea would be to divide through by the wave function in the wave equation, and we would then be left with the energy-eigenvalue E as a function of the potential, which we know is even around the origin: -ℏ^2/2m (d^2 ψ(x))/(dx^2 )+V(x)ψ(x)=ψ(x)E. Dividing by ψ(x): -ℏ^2/2m (d^2)/(dx^2 )+V(x)=E And the eigenvalue is now a function of the potential only. Is this an illegal move? Or where do I go from here?
Yes, that's an illegal move! What do you think "d^2/x^2", without a function, even means? d^2ψ(x)/dx is NOT "d^2/dx^2" multiplied by
ψ(x), it is the operator applied to the function. You cannot just "divide" by ψ(x).

Can you use the "chain rule" to determine dψ(-x)/dx and d^2ψ(-x)/dx^2?  Tags eigenvalue, equation, one-dimension, psix, schrodinger, show, solution, symmetry, time-independent Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Physics Forum Discussions Thread Thread Starter Forum Replies Last Post Instantonly Quantum Physics 17 Sep 2nd 2016 07:23 AM pre pt marc Electricity and Magnetism 1 Sep 25th 2011 08:43 PM aminr Advanced Mechanics 3 May 27th 2010 04:22 PM transgalactic Advanced Mechanics 0 Feb 6th 2009 06:55 AM SengNee General Physics 1 May 31st 2008 07:43 AM