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It's been a while since I've had DE, and it seems I have forgotten something basic, yet crucial. In solving the Time-Independent Schrodinger Wave Equation in one dimension, I have the following:

[tex]\left( -\frac{\hbar}{2m} \frac{\partial^2}{\partial x^2}+V\right) \Psi(x)=E\Psi(X)[/tex]

[tex]\frac{\partial^2}{\partial x^2}\Psi(x)+K^2\Psi(x)=0[/tex]

where V=0 and

[tex] K^2=\frac{2mE}{\hbar^2}[/tex]

And my question is: are the following solutions equal?

[tex]\Psi(x)=C_1e^{iKx}+C_2e^{-iKx}[/tex]

[tex]\Psi(x)=C_1Cos(Kx)+C_2Sin(Kx)[/tex]

I know that Euler's Formula is [tex]e^{i\theta x}=Cos(\theta x)+iSin(\theta x)[/tex], however inserting this into the first solution above does not result in the second. Thanks!

IHateMayonnaise

[tex]\left( -\frac{\hbar}{2m} \frac{\partial^2}{\partial x^2}+V\right) \Psi(x)=E\Psi(X)[/tex]

[tex]\frac{\partial^2}{\partial x^2}\Psi(x)+K^2\Psi(x)=0[/tex]

where V=0 and

[tex] K^2=\frac{2mE}{\hbar^2}[/tex]

And my question is: are the following solutions equal?

[tex]\Psi(x)=C_1e^{iKx}+C_2e^{-iKx}[/tex]

[tex]\Psi(x)=C_1Cos(Kx)+C_2Sin(Kx)[/tex]

I know that Euler's Formula is [tex]e^{i\theta x}=Cos(\theta x)+iSin(\theta x)[/tex], however inserting this into the first solution above does not result in the second. Thanks!

IHateMayonnaise

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