Identifying the Potential and Eigenfunction for Lowest Energy

In summary, the eigenfunction shown corresponds to the second allowed energy, E2. It has a peak in the range of energies where discretely separated allowed energy states exist, and a peak in the range of energies where continously distributed allowed energy states exist.
  • #1
justanovice
2
0

Homework Statement


Consider the eigenfunction in the top part of the figure.
(a) Which of the three potentials illustrated in the bottom part of the
figure could lead to such an eigenfunction? Give qualitative
arguments to justify your answer. (b) The eigenfunction shown
is not the one corresponding to the lowest allowed energy for the
potential. Sketch the form of the eigenfunction which does
correspond to the lowest allowed energy E1. (c) Indicate on
another sketch the range of energies where you would expect
discretely separated allowed energy states, and the range of
energies where you would expect the allowed energies to be
continuously distributed. (d) Sketch the form of the eigenfunction which corresponds to
the second allowed energy E2. (e) To which energy level does the eigenfunction presented
on top of the figure correspond

I've attached the figure, but It might be easier to read in this file i found on the internet (problem #5): http://www.phys.ncku.edu.tw/~ccheng/MP2011/HW2_SE.pdf

The Attempt at a Solution


I have already done part (a) by identifying the sign of ψ and (V-E). I concluded that the correct potential is the last one on the bottom. You can see my solution in the attachment.
For part (b), i think the sketch should have two peaks, one in each classically allowed region (B/C and E/F in my drawing) since this is the minimum amount of oscillations that will work.
However, this makes me think the answer to part (e) is energy level two.
Which makes part (d) confusing.
Also, I'm not sure what part (c) means by "discretely separated allowed allowed energy states" and continuously distributed allowed energies.

Any help would be greatly appreciated.
 

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  • #2


Hm... I follow your reasoning about the wavefunction shown being the second energy level. I'm not sure what they're getting at there but I'll see if someone else can figure it out.

As for part (c), the problem is talking about bound states and scattering states. Or alternatively, think of it like this: can you identify certain significant energies at which the behavior of the corresponding eigenfunction changes? In other words, for some particular energy EA there is a qualitative difference between a wavefunction corresponding to E < EA and a wavefunction corresponding to E > EA. What energy is that? What is another energy EB at which the same could be said?
 
  • #3


thank you!, i understand part (c) now.

my final is tomorrow and i was told there would be a problem similar to this, so any insight on parts (d) and (e) would help.
 
  • #4


There are two possible wave functions that have one peak in the B/C and E/F. One is odd, and one is even.
 
  • #5
I think I have the same exact final as justanovice...exactly the same class...lol!
 

1. What is the significance of identifying the potential and eigenfunction for lowest energy?

Identifying the potential and eigenfunction for lowest energy is crucial in understanding the behavior and properties of a physical system. It allows us to determine the lowest possible energy state of a system and predict its behavior under different conditions.

2. How is the potential for lowest energy determined in a physical system?

The potential for lowest energy is determined through mathematical calculations and experiments. The most commonly used method is the Schrödinger equation, which describes the behavior of quantum particles in a potential field. Experiments also play a crucial role in determining the potential for lowest energy in real-world systems.

3. What is an eigenfunction and how is it related to the lowest energy state?

An eigenfunction is a mathematical function that represents a physical state in a system. In the context of identifying the potential and eigenfunction for lowest energy, the eigenfunction is the mathematical representation of the lowest energy state of a system. It is also known as the ground state.

4. Can the potential and eigenfunction for lowest energy change in a system?

Yes, the potential and eigenfunction for lowest energy can change in a system depending on the external conditions and interactions with other particles. For example, applying an external force or changing the temperature can alter the potential and eigenfunction for lowest energy in a physical system.

5. How does identifying the potential and eigenfunction for lowest energy contribute to scientific research?

Identifying the potential and eigenfunction for lowest energy is essential in various fields of science, including physics, chemistry, and materials science. It helps researchers understand the fundamental properties of matter and develop new technologies based on quantum mechanics. Additionally, it allows us to make accurate predictions about the behavior of complex systems and design experiments to test our theories.

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