Find the possible total energies (Quantum Physics)

In summary, the conversation discusses a problem in quantum physics involving two particles connected by a spring, with the total momentum and possible total energies being the main focus. The solution for the case of two different particles is discussed, and then the conversation moves on to the problem of two identical fermions. The expert suggests using a coordinate transformation to solve the problem and provides a resource for reference.
  • #1
BookWei
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Homework Statement


I'm doing problems for practice in quantum physics.
Consider two particles of the mass m in one dimension with coordinates being denoted by x and they are
connected by a spring with spring constant k. Suppose that the total momentum of the system is p.
Find all possible total energies for the following cases :
(1)two particles are different (2)two particles are identical fermions.

2. The attempt at a solution
(1) I try to guess the answer...
Total energy is the sum of potential and kinetic energy. Now our particles have the same mass and they are one-dimensional. Moreover, they are non-identical. Now potential energy is based on spring constant K therefore V=1/2*K*x^2 . Now considering the harmonic oscillator in classic sense total energy E= T +V
Therefore E = P^2/2m + 1/2*mω^2*x^2.

considering energy from quantum mechanical point of view, we know P= -iℏ d/dx =p' and x=x' hamiltonian becomes, H= 1/2 p'^2/2m + 1/2*mω'^2*x^2
now considering the particles time independent
H'ψ(x) = Eψ(x)
the eigenvalues of this Hamiltonian is based on En = (n+ 1/2)ℏω where ground state has non-zero energy.

(2) I have no idea how to start this problem.

Thank you for your help.
 
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  • #2
BookWei said:
Thank you for your help.

I'm not sure at all how to interpret this problem. My first thought was that you have two particles in a harmonic oscillator, but I suspect that is not what is intended.

I'm not sure how you can have two fermions connected by a spring. How could you attach a fermion to a spring?

Perhaps what is meant is simply to calculate the potential for a classical system of this type and then translate that to a quantum potential - based on particle repulsion and attraction, rather than a spring! - and solve the resulting Schrodinger equation.

In any case, your solution to 1), which appears to be simply a solution to a single particle SHO, cannot be correct.
 
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  • #3
Thanks a lot.
I upload the original problem file.

I will try to solve the problem (1).
 

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  • #4
BookWei said:
Thanks a lot.
I upload the original problem file.

I will try to solve the problem (1).

Okay, so the question setter does believe you can attach fermions and bosons to a spring. On the face of it, the natural length of the spring should be relevant, but I think you'll just have to do the maths and see what happens.
 
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  • #5
BookWei said:
H'ψ(x) = Eψ(x)
You have a two-particle system. So, you can consider the wavefunction to be a function of two position coordinates: ψ(x1, x2).

Do you know how to do a coordinate transformation which separates out the center-of-mass motion and the motion relative to the center of mass (with a reduced mass μ)?
 
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  • #6
I tried to solve this problem for two days.
But I still do not know how to solve it...
 
  • #7
I suggest that you change variables from ##x_1## and ##x_2## to ##X_c## and ##x##, where ##X_c## is the coordinate of the center of mass and ##x = x_2 - x_1##. This is a standard method for reducing the two-body problem to two independent one-body problems.

Hopefully you've seen this before. See the first 3 or 4 pages here: http://physics.oregonstate.edu/~corinne/COURSES/ph426/notes2.pdf
 
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  • #8
TSny said:
I suggest that you change variables from ##x_1## and ##x_2## to ##X_c## and ##x##, where ##X_c## is the coordinate of the center of mass and ##x = x_2 - x_1##. This is a standard method for reducing the two-body problem to two independent one-body problems.

Hopefully you've seen this before. See the first 3 or 4 pages here: http://physics.oregonstate.edu/~corinne/COURSES/ph426/notes2.pdf
Thank you so much !
 

FAQ: Find the possible total energies (Quantum Physics)

What is the concept of total energy in quantum physics?

The concept of total energy in quantum physics refers to the sum of potential energy and kinetic energy of a particle or system. It takes into account both the particle's position and its motion, and is a fundamental aspect of understanding the behavior of quantum systems.

How do you calculate the total energy of a quantum system?

The total energy of a quantum system can be calculated using the Schrodinger equation, which takes into account the Hamiltonian operator, the wave function, and the time-dependent Schrodinger equation. This calculation allows for the prediction of the total energy of a system at a specific time and location.

Can the total energy of a quantum system change over time?

Yes, the total energy of a quantum system can change over time. This is because quantum systems are dynamic and can undergo changes in their potential energy and kinetic energy, resulting in a change in their total energy. However, the total energy of a closed quantum system remains constant over time.

How does the Heisenberg uncertainty principle relate to total energy in quantum physics?

The Heisenberg uncertainty principle states that it is impossible to know both the position and the momentum of a particle with absolute certainty. This means that the total energy of a quantum system can never be precisely determined, as it is dependent on both position and momentum. Therefore, the uncertainty principle plays a crucial role in understanding the behavior of quantum systems and their total energy.

What is the significance of total energy in quantum physics?

The total energy of a quantum system is significant because it determines the behavior and characteristics of the system. It allows for the prediction of the system's dynamics and helps in understanding the fundamental principles of quantum mechanics. Furthermore, the concept of total energy is essential in various applications of quantum physics, such as quantum computing and quantum cryptography.

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