# Particle in XY Plane Homework: Separating Variables

• gfd43tg
In summary: This is what I was trying to do, but I don't know how to do it. "I don't know how to do it" is not an answer to the question.
gfd43tg
Gold Member

## The Attempt at a Solution

a) I know that I am allowed to separate variables, such that ##\psi (r, \theta) = R(r) \Theta(\theta)##, but the part with the exponential is yet to be seen. So I put it into the Hamiltonian,
$$H = - \frac {\hbar^{2}}{2 \mu} \Big ( \frac {1}{R} \frac {\partial^{2} R}{\partial r^{2}} + \frac {1}{R} \frac {1}{r} \frac {\partial R}{\partial r} + \frac {1}{r^{2}} \frac {1}{\Theta} \frac {\partial^{2} \Theta}{\partial \theta^{2}} \Big ) + V$$

Since V is a function of r, it is not a constant, so I can't just call every term with derivatives as constants, can I? This is what I would normally be able to do, particularly if V = 0. So how do I proceed from here?

You can take another angular derivative, then all the non-trivial R terms will go away.
The problem statement is misleading, there are solutions that cannot be written in that way, e.g. superpositions of states with different m.

What is the other angular derivative you are talking about? I just see the one in my equation.

"Another" as in "one more". You can calculate ##\frac{dH}{d\theta}##.

Wouldn't that give some sort of triple derivative on the right side of my equation for the third term in the parenthesis?

Sure.

$$-\frac{\hbar^2}{2\mu}\left(\frac{R''}{R} + \frac{R'}{rR} + \frac{\Theta'}{r^2\Theta}\right) + V = E.$$ Multiply by ##r^2## and you can separate the terms into those depending on ##r## and those that depend only on ##\theta##.

Okay, so I can separate the equation

$$-(H-V)r^{2} \frac {2 \mu}{\hbar^{2}} - \frac {r^{2}}{R} R'' - \frac {r}{R} R' = \frac {\Theta''}{\Theta}$$
But how can I solve this?

The left side is independent of theta, which means the whole equation has to have the same value for all values of theta. You can set the left side equal to some constant (it will depend on r but not on theta). This is the same as setting the derivative (with respect to theta) to zero.

## 1. What is a "particle in XY plane"?

A particle in XY plane refers to a particle that moves in a two-dimensional coordinate system, where the X-axis represents horizontal distance and the Y-axis represents vertical distance.

## 2. What is the concept of "separating variables" in this context?

"Separating variables" refers to a mathematical technique used to solve differential equations for a particle in XY plane. It involves isolating the variables on each side of the equation to solve for the individual variables separately.

## 3. What are the applications of studying "particle in XY plane" and "separating variables"?

The study of particle in XY plane and separating variables has various real-world applications, such as predicting the position and movement of objects in space, analyzing the motion of projectiles, and understanding the behavior of fluids and gases.

## 4. How does one approach solving a "particle in XY plane" homework problem?

To solve a "particle in XY plane" homework problem, you should first identify the variables given in the problem and the differential equation that represents the motion of the particle. Then, you can use the technique of separating variables to solve the equation and find the solution for the position of the particle at a given time.

## 5. What are some tips for successfully completing a "particle in XY plane" homework assignment?

Some tips for successfully completing a "particle in XY plane" homework assignment include understanding the concept of separating variables, practicing solving similar problems, and seeking help from your teacher or peers if you encounter any difficulties. It is also important to carefully read and understand the given problem, and show all your steps and calculations clearly.

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