Finding the quantized energies of a particle

In summary, the conversation discusses the attempt to solve for the quantized energies of a particle in a potential with a constant b. The individual is following a derivation from Griffiths' Quantum textbook and uses a substitution and power series expansion to find the energy values, but struggles to find the correct recurrence relations. They also mention the need for a regular solution at the origin.
  • #1
physics148
13
0

Homework Statement



Okay, so the question I'm trying to solve is to find the quantized energies for a particle in the potential:

$$V(x)=V_0 \left ( \frac{b}{x}-\frac{x}{b} \right )^2$$

for some constant b.

The Attempt at a Solution


I am following along with the derivation of the quantized energies for the SHO in griffiths quantum(2nd ed) from pages 51-56. I used the schrodinger equation with the substitution $$\psi=\frac{\tilde \psi}{\sqrt{x}}$$ as well as $\tilde x=bx$(to make it unitless) and substituted it into the schrodinger equation and obtained after much simplification
$$ \tilde \psi''- \tilde \psi' x^{-1}+ \tilde \psi (\frac{3/4+V_0}{x^2}+V_0x^2-(E+2V_0)=0 $$

Then, I followed along with the derivation of Griffiths, by finding an integrating factor and trying to expand $\psi$ as a power series which led me to the following
$$\psi(\tilde x)=f(x)e^-\frac{\tilde x^2}{2} \quad \text{Eq 2.77 in Griffiths 2nd ed}$$
$$f(x)=\sum_{n=0}^{\infty}C_nx^n$$
$$f'(x)=\sum_{n=0}^{\infty}nC_nx^{n-1}$$
$$f''(x)=\sum_{n=0}^{\infty}n(n-1)C_nx^{n-2}$$

This of course leads to(dropping the tilde's on the x for simplicity) $$\tilde \psi= f( x)e^{- x^2/2}$$
$$\tilde \psi '=e^{- x^2/2} (f'(x)-xf(x)) $$
$$\tilde \psi ''=e^{-x^2/2}(f''(x)-2xf'(x)+f(x)(x^2-1))$$

I believe these are the correct $\psi$'s however my problem now is when I sub them back into the schrodinger equation I found above, I don't get the same cancellation that happens for the SHO, so I am struggling to find the recurrence relations in order to find the quantized energies. If anyone can help me find the energies I would be much appreciative.
 
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  • #2
In your differential equation you have a term proportional to ##1/x^2##. Your solution must be regular at the origin.
Therefore, you need to use an expansion of the form
$$f(x)=x^2\sum_{n=0}^{\infty}C_n x^n$$.

physics148 said:

Homework Statement



Okay, so the question I'm trying to solve is to find the quantized energies for a particle in the potential:

$$V(x)=V_0 \left ( \frac{b}{x}-\frac{x}{b} \right )^2$$

for some constant b.

The Attempt at a Solution


I am following along with the derivation of the quantized energies for the SHO in griffiths quantum(2nd ed) from pages 51-56. I used the schrodinger equation with the substitution $$\psi=\frac{\tilde \psi}{\sqrt{x}}$$ as well as $\tilde x=bx$(to make it unitless) and substituted it into the schrodinger equation and obtained after much simplification
$$ \tilde \psi''- \tilde \psi' x^{-1}+ \tilde \psi (\frac{3/4+V_0}{x^2}+V_0x^2-(E+2V_0)=0 $$

Then, I followed along with the derivation of Griffiths, by finding an integrating factor and trying to expand $\psi$ as a power series which led me to the following
$$\psi(\tilde x)=f(x)e^-\frac{\tilde x^2}{2} \quad \text{Eq 2.77 in Griffiths 2nd ed}$$
$$f(x)=\sum_{n=0}^{\infty}C_nx^n$$
$$f'(x)=\sum_{n=0}^{\infty}nC_nx^{n-1}$$
$$f''(x)=\sum_{n=0}^{\infty}n(n-1)C_nx^{n-2}$$

This of course leads to(dropping the tilde's on the x for simplicity) $$\tilde \psi= f( x)e^{- x^2/2}$$
$$\tilde \psi '=e^{- x^2/2} (f'(x)-xf(x)) $$
$$\tilde \psi ''=e^{-x^2/2}(f''(x)-2xf'(x)+f(x)(x^2-1))$$

I believe these are the correct $\psi$'s however my problem now is when I sub them back into the schrodinger equation I found above, I don't get the same cancellation that happens for the SHO, so I am struggling to find the recurrence relations in order to find the quantized energies. If anyone can help me find the energies I would be much appreciative.
 

Related to Finding the quantized energies of a particle

1. How do you determine the quantized energies of a particle?

The quantized energies of a particle can be determined through a process called quantum mechanics. This involves using mathematical equations and principles to calculate the energy levels of a particle based on its properties and the surrounding environment.

2. What factors affect the quantized energies of a particle?

The quantized energies of a particle are affected by several factors, including the particle's mass, velocity, and position. The type of particle also plays a role, as different particles have different energy levels. Additionally, external forces or fields can influence the quantized energies of a particle.

3. Can the quantized energies of a particle change?

Yes, the quantized energies of a particle can change. This can occur through interactions with other particles or through changes in the particle's properties. For example, if a particle gains or loses energy, its quantized energies will also change.

4. How do scientists use the quantized energies of particles in research?

The quantized energies of particles are used in a variety of scientific research, particularly in the field of quantum mechanics. They help scientists understand the behavior and properties of particles, as well as how they interact with each other and their environment. This information is crucial in fields such as physics, chemistry, and engineering.

5. What are the practical applications of understanding the quantized energies of particles?

Understanding the quantized energies of particles has many practical applications, such as in the development of new technologies and materials. It also plays a role in fields such as energy production, medical imaging, and cryptography. The study of quantized energies also helps advance our understanding of the fundamental laws of the universe.

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