Is the Potential in Quantum Hamiltonian II Always Real?

In summary, the conversation discusses the Hamiltonian of a system and the question of whether the potential, denoted by V, is real or complex. The speaker presents a proof using normalized Eigenfunctions to show that the potential must be real if the expected value of the kinetic energy is always real. The other speaker raises a question about the relationship between the potential and the energy, and the first speaker responds by noting that the energy is an observable and its operator, the Hamiltonian, is hermitian which implies real eigenvalues.
  • #1
eljose
492
0
Let be a Hamiltonian in the form H=T+V we don,t know if V is real or complex..all we know is that if E_n is an energy also E*_n=E_k will be another energy, my question is if this would imply V is real...

my proof is taking normalized Eigenfunctions we would have that:

[tex](<\phi_{n}|H|\phi_{n}>)*=<\phi_{k}|H|\phi_{k}> [/tex]

so the expected value of T is always real,then we would have the identity with the complex part b(x) of the potential:

[tex]\int_{-\infty}^{\infty}dx(|\phi_{n}|^{2}+|\phi_{k}|^{2})b(x)=0 [/tex]

for every k,and n so necessarily b=0 so the potential is real and all the eigenfunctions would be real.
 
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  • #2
I find the question a bit weird. The energy of a system is an observable and its corresponding operator is the Hamiltonian and is thus hermitian which implies having real eigenvalues. If you take V nonreal H isn't Hermitian anymore.
 
  • #3


I would respond by saying that your proof is valid and provides evidence that the potential in the Quantum Hamiltonian II is indeed real. However, it is important to note that this proof is based on certain assumptions and may not be applicable in all cases. Further research and experimentation may be needed to fully understand the nature of the potential in the Quantum Hamiltonian II. Additionally, it is important to consider the implications of this finding and how it may impact our understanding of quantum mechanics and its applications.
 

1. What is a Quantum Hamiltonian II?

A Quantum Hamiltonian II is a mathematical operator that describes the total energy and motion of a quantum system.

2. How is Quantum Hamiltonian II different from the original Quantum Hamiltonian?

Quantum Hamiltonian II is an extension of the original Quantum Hamiltonian, which takes into account additional factors such as spin and relativity, making it a more accurate representation of quantum systems.

3. What is the importance of Quantum Hamiltonian II in quantum mechanics?

Quantum Hamiltonian II is essential in solving the Schrodinger equation, which is a fundamental equation in quantum mechanics that describes how quantum systems evolve over time.

4. Are there any real-world applications of Quantum Hamiltonian II?

Yes, Quantum Hamiltonian II has various applications in fields such as quantum computing, quantum chemistry, and quantum field theory. It is used to model and understand complex quantum systems and phenomena.

5. Is Quantum Hamiltonian II a widely accepted concept in the scientific community?

Yes, Quantum Hamiltonian II is a well-established concept in quantum mechanics and is widely accepted by the scientific community. It has been extensively studied and confirmed through various experiments and applications.

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