Solution of Quantum differential equation

In summary, the conversation is about finding the correct answer for the equation φ=Asin(kx)+Bcos(kx), where the constants A and B can be either real or complex. One person believes that if φ is complex valued, then the general solution is as the first answer says with complex constants A and B, while the other person believes that if φ is real valued, then the correct answer is as the answer key says with real constants A and B. The conversation concludes with discussing how to get from the general answer to the specific answer for real valued φ.
  • #1
Edge5
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pZgfb3s

HPKZ6KD.jpg

(I think I couldn't add the image)
you can see my answer in link

https://pasteboard.co/HPKZ6KD.jpg

(Please first see my answer in the link)
But in answer it is φ= Asin(kx) + Bcos(kx)

I know that euler formula is eix = cosx +isinx

But I can't get this answer can you help me?
 
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  • #2
In the answer key does it say if the constants A,B are real or complex?

My opinion is that if ##\phi## is complex valued then the general solution is as your answer says (and in your answer the constants A,B can be complex constants).
However if ##\phi## is real valued then the correct answer is as the answer key says that is ##\phi=A\sin(kx)+B\cos(kx)## where A,B are real constants here.
 
  • #3
Delta2 said:
In the answer key does it say if the constants A,B are real or complex?

My opinion is that if ##\phi## is complex valued then the general solution is as your answer says (and in your answer the constants A,B can be complex constants).
However if ##\phi## is real valued then the correct answer is as the answer key says that is ##\phi=A\sin(kx)+B\cos(kx)## where A,B are real constants here.
Assuming answer is real. How do I get from this general answer to ##\phi=A\sin(kx)+B\cos(kx)##
 
  • #4
if we assume ##\phi## is real valued then from your general answer (for which i ll use ##A'## and ##B'## to denote the complex constants as well as ##\phi'## for the complex valued function) we ll have (i use Euler's formula to rewrite your general answer).

$$\phi'=A'\cos(kx)+B'\cos(-kx)+i(A'\sin(kx)+B'\sin(-kx))=(A'+B')\cos(kx)+i(A'-B')\sin(kx) \text{(1)}$$

So we just looking for complex constants ##A',B'## such that ##(A'+B')=B (2) ## and ##(A'-B')=-iA (3)## and then for these constants it would be ##\phi=\phi'##. The system of equations (2),(3) has unique solutions for ##A',B'## given that ##A,B## are real.
 
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  • #5
Now that i think of it again, if we allowed for ##\phi## to be complex valued (since it is a wave function it would be complex valued in general case), and also allow for constants A,B to be complex in the answer key (the answer that your book says), then the answer key and your answer are equivalent.
 
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Related to Solution of Quantum differential equation

1. What is a quantum differential equation?

A quantum differential equation is a mathematical expression that describes the behavior of quantum systems, which are systems that follow the principles of quantum mechanics. These equations are used to calculate the probability of a particle's behavior or the evolution of a quantum state over time.

2. How is a quantum differential equation solved?

A quantum differential equation is solved by using mathematical techniques such as separation of variables, series solutions, or numerical methods like the Runge-Kutta method. These methods involve finding the eigenvalues and eigenfunctions of the equation, which represent the possible solutions.

3. What is the significance of finding the solution to a quantum differential equation?

The solution to a quantum differential equation is significant because it allows us to make predictions about the behavior of quantum systems. By solving the equation, we can determine the probabilities of different outcomes and understand how the system evolves over time.

4. Can a quantum differential equation have multiple solutions?

Yes, a quantum differential equation can have multiple solutions. In fact, many quantum systems have a continuous spectrum of solutions, which means that there are an infinite number of possible solutions that can describe the behavior of the system.

5. What are some real-world applications of solving quantum differential equations?

Solving quantum differential equations has many practical applications, such as in quantum chemistry, quantum computing, and quantum mechanics. It allows us to understand and predict the behavior of particles at the quantum level, which is crucial for advancements in technology and scientific research.

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