Restrictions on Numbers in Hermitian Hamiltonian Equation

In summary, the conversation discusses the requirement for a Hamiltonian to be Hermitian and the use of ladder operators to write it in a specific form. The first part concludes that the numbers a and b must be real in order for H to be Hermitian. The second part suggests using an operator B to rewrite H in the form of u B*B + const and determines the spectrum of H through this method. One participant also asks for clarification on the specific ladder operators being referred to.
  • #1
bon
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Homework Statement



In terms of the usual ladder operators A, A* (where A* is A dagger), a Hamiltonian can be written H = a A*A + b(A + A*)

What restrictions on the values of the numbers a and b follow from the requirement that H has to be Hermitian?

Show that for a suitably chosen operator B, H can be written

H = u B*B + const.

where [B,B*] = 1. Hence determine the spectrum of H


Homework Equations





The Attempt at a Solution



So i think the answer for the first part is that both a and b must be real/

Not sure about the next part though, how do i show this? and then how do i work out the spectrum of H?

Thanks
 
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  • #2


ideas?
 
  • #3


i guess you are meant to write B in terms of A and A*, right? I just don't see how to do it though... also don't see how it helps you determine the spectrum of H...! Thanks
 
  • #4


hello I found your problem today. Can you explain better what ladder operator are you referring to. I know two one in the harmonic osillator and another in angular momentum
 
  • #5

What is a Hamiltonian in physics?

A Hamiltonian in physics is a mathematical operator that represents the total energy of a system. It is used in classical mechanics and quantum mechanics to describe the dynamics of a physical system.

How is a Hamiltonian different from a Lagrangian?

A Hamiltonian and a Lagrangian are both mathematical functions used to describe the dynamics of a physical system. However, the Hamiltonian is a function of both position and momentum, while the Lagrangian is only a function of position. Additionally, the Hamiltonian is used in Hamiltonian mechanics, while the Lagrangian is used in Lagrangian mechanics.

What is the Hamiltonian in quantum mechanics?

In quantum mechanics, the Hamiltonian is a mathematical operator that represents the total energy of a quantum system. It includes the kinetic energy and potential energy of the system, and its eigenvalues correspond to the possible energy levels of the system.

How do you solve Hamiltonian equations?

The Hamiltonian equations of motion can be solved using Hamilton's equations, which relate the derivatives of the position and momentum variables to the Hamiltonian function. These equations can be solved analytically or numerically to determine the evolution of a physical system over time.

What are some practical applications of Hamiltonian mechanics?

Hamiltonian mechanics has many practical applications in physics, engineering, and other fields. It is used to study the motion of particles in a magnetic field, the dynamics of a pendulum, and the behavior of complex systems such as galaxies. It is also used in quantum mechanics to study the behavior of atoms and molecules.

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