Spin-3/2 particle and degeneracy in excited state

In summary, the question is asking for the minimum energy of a particle with mass m in a 1D potential, using the uncertainty relation. The solution involves using a constant with the dimension of length and leads to the conclusion that the minimum energy is a multiple of (m \hbar^2 g^2)^{\frac 1 3}. Further examples of using the uncertainty principle to estimate ground state energies can be found online.
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ShayanJ
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Homework Statement



Consider a particle with mass m in the following 1D potential:
[itex]
V(x)=\left\{ \begin{array}{lr} mgx \ \ \ x>0 \\ \infty \ \ \ \ \ \ \ x\leq 0 \end{array} \right.
[/itex]
What is its minimum energy calculated using the uncertainty relation?

Homework Equations



[itex]
\Delta x \Delta p \geq \frac{\hbar}{2}
[/itex]

The Attempt at a Solution



My problem is, I don't know what to use for [itex] \Delta x [/itex]! I see no length scale I can use. The only thing that came into my mind was making a constant with the dimension of length using the available constants and so I got [itex] \alpha(\frac{\hbar^2}{gm^2})^{\frac 1 3} [/itex](where [itex] \alpha[/itex] is a dimensionless constant) and this gives [itex] \Delta p \geq \frac{1}{2\alpha} (m^2 \hbar g)^{\frac 1 3} \Rightarrow E_0=\frac{1}{8\alpha^2} (m \hbar^2 g^2)^{\frac 1 3}[/itex].
But the problem is, this method can give any multiple of [itex] (m \hbar^2 g^2)^{\frac 1 3}[/itex]!
What should I do?
Thanks
 
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  • #3
TSny said:
I'm not sure exactly what the questioner had in mind. But it might be similar to the fairly well-known example of using the uncertainty principle to estimate the ground state energy of the hydrogen atom.

See

http://quantummechanics.ucsd.edu/ph130a/130_notes/node98.html

http://www.pha.jhu.edu/~rt19/hydro/node1.html

http://www.uio.no/studier/emner/matnat/astro/AST1100/h06/undervisningsmateriale/lecture-2.pdf

Thanks man, it was very helpful.
 
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1. What is a spin-3/2 particle?

A spin-3/2 particle is a type of elementary particle that has a spin quantum number of 3/2, which is a property that describes the intrinsic angular momentum of the particle. This type of particle is typically found in the nuclei of atoms and is an important factor in nuclear physics and quantum mechanics.

2. How does degeneracy occur in excited states of spin-3/2 particles?

Degeneracy occurs in excited states of spin-3/2 particles when there are multiple energy levels that have the same energy. This can happen when the particle's spin interacts with other particles or external forces, causing the energy levels to shift and overlap. Degeneracy is an important concept in understanding the behavior and characteristics of spin-3/2 particles.

3. Can spin-3/2 particles have different orientations of their spin?

Yes, spin-3/2 particles can have four different orientations of their spin: up, down, left, and right. These orientations correspond to the four possible values of the spin quantum number, which is a half-integer value that describes the direction and magnitude of the particle's spin.

4. How is the degeneracy of spin-3/2 particles related to their multiplicity?

The degeneracy of spin-3/2 particles is related to their multiplicity, which is the number of possible states that the particle can occupy. In general, the higher the degeneracy, the higher the multiplicity, as more energy levels and spin orientations are available to the particle.

5. Why is the concept of degeneracy important in studying spin-3/2 particles?

The concept of degeneracy is important in studying spin-3/2 particles because it provides insight into the behavior and properties of these particles. In particular, the degeneracy of energy levels and spin orientations can affect the stability, interactions, and transitions of spin-3/2 particles, making it a crucial factor in understanding their behavior in various physical systems.

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