Schrodinger equation reduction using substitution

In summary: Therefore, the Schrodinger equation has been successfully reduced using the substitution $w= A^{1/3} (x - \frac{B}{A})$.
  • #1
phagist_
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SOLVED: Schrodinger equation reduction using substitution

Homework Statement


Given

[tex] \frac{d^2 \psi}{dx^2} - Ax\psi + B\psi = 0[/tex]

make a substitution using
[tex]w= A^{1/3} (x - \frac{B}{A})[/tex]

to get
[tex] \frac{d^2 \psi}{dw^2} - w\psi = 0[/tex]

Homework Equations





The Attempt at a Solution


I use

[tex] \frac{d\psi}{dx} = \frac{d\psi}{dw} \frac{dw}{dx}[/tex]

then
[tex]\frac{d^2\psi}{dx^2} = \frac{d}{dx}[ \frac{d\psi}{dw} \frac{dw}{dx}][/tex]

then

[tex]\frac{d^2\psi}{dx^2} = \frac{d\psi}{dwdx} \frac{dw}{dx} + \frac{d^2w}{dx^2} \frac{d\psi}{dw}[/tex]

but the [tex]\frac{d^2w}{dx^2}[/tex] equals zero, since x is linear in w.

which implies

[tex]\frac{d^2\psi}{dx^2} = \frac{d\psi}{dwdx} \frac{dw}{dx}[/tex]

and [tex]\frac{dw}{dx} = A^{1/3}[/tex]

but I'm not sure how to evaluate the

[tex]\frac{d\psi}{dwdx}[/tex] term (I'm not sure If it should even be there.. did I use the chain rule correctly?)

Then I'll sub in [tex]\frac{d\psi}{dwdx} A^{1/3}[/tex] for [tex] \frac{d^2 \psi}{dx^2}[/tex] and hopefully it all works out.

Any help would be greatly appreciated.

Edit: SOLVED
 
Last edited:
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  • #2
:\frac{d^2 \psi}{dx^2} - Ax\psi + B\psi = 0make a substitution using w= A^{1/3} (x - \frac{B}{A})to get \frac{d^2 \psi}{dw^2} - w\psi = 0Solution:Using the chain rule, we have\frac{d^2 \psi}{dx^2} = \frac{d^2 \psi}{dw^2} \left(\frac{dw}{dx}\right)^2 + \frac{d\psi}{dw} \frac{d^2w}{dx^2}Substituting in our expression for $w$, we have\frac{d^2 \psi}{dx^2} = \frac{d^2 \psi}{dw^2} \left(A^{1/3}\right)^2 + \frac{d\psi}{dw} \frac{d^2w}{dx^2}The second term is zero, since $x$ is linearly related to $w$. Thus, we have\frac{d^2 \psi}{dx^2} = \frac{d^2 \psi}{dw^2} A^{2/3}Substituting this into the original equation, we get\frac{d^2 \psi}{dw^2} A^{2/3} - Ax\psi + B\psi = 0Multiplying both sides by $A^{-2/3}$, we get\frac{d^2 \psi}{dw^2} - w\psi = 0,which is the desired result.
 
  • #3
!

Thank you for sharing your attempt at solving the Schrodinger equation reduction using substitution problem. Your approach seems to be on the right track. However, there are a few corrections and clarifications that can be made.

Firstly, when you use the chain rule to find \frac{d^2\psi}{dx^2}, you should have two terms. One term will have \frac{d\psi}{dw} and the other term will have \frac{d^2\psi}{dw^2}. This is because when you differentiate \frac{d\psi}{dw} with respect to x, you will need to use the chain rule again.

Secondly, when you substitute \frac{d^2\psi}{dx^2} = \frac{d\psi}{dwdx} \frac{dw}{dx} into the original equation, you will also need to substitute w = A^{1/3} (x - \frac{B}{A}) into the equation.

Lastly, you can evaluate \frac{d\psi}{dwdx} by using the fact that w = A^{1/3} (x - \frac{B}{A}). This means that \frac{d\psi}{dwdx} = \frac{d\psi}{dw} \frac{dw}{dx} = A^{1/3} \frac{d\psi}{dw}.

I hope this helps you to solve the problem. Keep in mind that the Schrodinger equation is a fundamental equation in quantum mechanics and has many applications in understanding the behavior of particles at the microscopic level. Good luck!
 

1. What is Schrodinger equation reduction using substitution?

Schrodinger equation reduction using substitution is a method used in quantum mechanics to simplify the Schrodinger equation by substituting a variable with a new one. This allows for the equation to be solved more easily, making it a useful tool in understanding and predicting the behavior of quantum systems.

2. How does substitution simplify the Schrodinger equation?

By substituting a variable in the Schrodinger equation with a new one, the equation becomes more manageable and easier to solve. This is because the new variable may have different properties or values that make the equation simpler to manipulate.

3. What is the significance of the Schrodinger equation in quantum mechanics?

The Schrodinger equation is a fundamental equation in quantum mechanics that describes the behavior of quantum particles, such as electrons, in a given system. It allows us to calculate the probability of finding a particle in a certain location or state, and is essential in understanding the behavior of matter at the atomic and subatomic level.

4. Can substitution be used for any variable in the Schrodinger equation?

Yes, substitution can be used for any variable in the Schrodinger equation, as long as it does not change the physical meaning of the equation. It is important to choose the appropriate substitution that simplifies the equation without altering its fundamental properties.

5. Are there any limitations to using substitution in reducing the Schrodinger equation?

While substitution can be a useful tool in simplifying the Schrodinger equation, it may not always be possible to find a suitable substitution. In some cases, the substitution may lead to a more complicated equation or may not be physically meaningful. Therefore, it is important to carefully consider the choice of substitution and its implications.

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