Nonreal solutions to the TISE

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In summary, the conversation discusses the expression of a non-real solution to the Schrodinger equation, \psi(x), as a linear combination of real solutions. It is suggested that this can be achieved by setting \psi(x) = \psi_1(x) +i\psi_2(x), where \psi_1,2(x) are real functions. Two approaches are mentioned, one involving plugging the real solutions into the Schrodinger equation and the other involving setting \psi(x) equal to a combination of real solutions. The conversation ends with a request for clarification or guidance on the right approach.
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G01
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1. I am try to show that a non-real solution to the shrodinger eq., [tex]\psi(x)[/tex] can be expressed as a linear combination of real solutions.
2. Here's what I know:
Shrodinger Eq. [tex] -\frac{\hbar^2}{2m}\frac{d^2\psi(x)}{dx^2} +V\psi(x) = E\psi(x)[/tex]

[tex]\psi[/tex] [tex] \psi*[/tex] are non-real solutions

[tex] (\psi + \psi*)[/tex] and[tex]i(\psi -\psi*)[/tex] are real solutions.


3. Well, to start, since I am given two real solutions, I am assuming [tex]\psi[/tex] can be expressed as a linear combination of the two real solutions above. I tried this two different ways:

First I tried plugging the real solutions into the shrodinger eq. and seeing where it got me. I was able to split the equation into two separate equations involving only the original psi and psi* but I don't think this helps me.

I also tried setting psi = A +Bi and seeing if I could combine the real solutions in a way so they equal psi. Also, this hasn't been of much help so far. If I am somewhat on track but am confused, please point me in the right direction. Or am I completely off track? Thank you for your insight.
 
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  • #2
Take [itex] \psi (x)= \psi_{1}(x)+i\psi_{2}(x) \ , \psi_{1,2}(x)\in \mathbb{R} [/itex]. Plug the psi in the SE and and separate the real and imaginary part. That's all.
 
  • #3
Thanks Dexter.
 

1. What are nonreal solutions to the TISE?

Nonreal solutions to the TISE refer to solutions that involve imaginary numbers (i) and complex numbers. These solutions arise when solving the Time-Independent Schrödinger Equation (TISE) for quantum mechanical systems.

2. Why do nonreal solutions occur in the TISE?

Nonreal solutions occur in the TISE because it represents the behavior of quantum mechanical systems, which can have both real and imaginary components. These solutions provide a more accurate understanding of the quantum world and allow for more precise predictions.

3. How do nonreal solutions affect the TISE?

Nonreal solutions play a crucial role in the TISE as they contribute to the overall wave function of a quantum system. They also provide information about the energy levels and behavior of the system.

4. How can I determine if the TISE has nonreal solutions?

To determine if the TISE has nonreal solutions, you can solve the equation and look for the presence of imaginary numbers (i) or complex numbers. These solutions will also appear in the wave function of the system.

5. What is the significance of nonreal solutions in quantum mechanics?

Nonreal solutions are significant in quantum mechanics as they provide a more accurate understanding of the behavior of quantum systems. They contribute to the overall wave function, energy levels, and observables of the system, allowing for more precise predictions and insights into the quantum world.

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