Hamiltonian Matrix Eq. 8.43 Explained - Feynman III Quantum Mechanics

In summary, the Hamiltonian matrix is a mathematical operator used in quantum mechanics to describe the total energy of a system. It is represented by a square matrix and is essential in solving the Schrödinger equation, calculating energy levels and transition probabilities, and describing interactions between particles and fields in Feynman III Quantum Mechanics. Eq. 8.43 in Feynman III Quantum Mechanics is a specific equation that represents the time-independent Schrödinger equation and is solved using the Hamiltonian matrix. The Hamiltonian matrix is also used to calculate amplitudes of particle interactions in Feynman diagrams. While the concept of a Hamiltonian matrix originated in classical mechanics, its application in quantum mechanics is fundamental and differs from its use
  • #1
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In the volume III of R Feynman series which is on Quantum Mechanics , please explain to me the eq.8.43 given on page 1529, i know how we got the equation but the 2nd part of 1st equation (H12)C2, what does it mean ?
 
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  • #2
This equation is simply an application of eq. 8.39 to a 2-state system. C1 and C2 are the components of the wave function along each of the 2 basis states. H12 and H21 are the off-diagonal terms in the Hamiltonian. [tex]

\mathcal{H} \equiv \left(
\begin{array}{cc}
H_{11} & H_{12}\\
H_{21} & H_{22}
\end{array}
\right)

[/tex]
 
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  • #3


The Hamiltonian matrix is a mathematical representation of the total energy of a quantum system, and it plays a crucial role in quantum mechanics. In equation 8.43, Feynman is discussing the Hamiltonian matrix for a system with two states, labeled as state 1 and state 2.

The first part of the equation, H11 and H22, represents the potential energy of the system in state 1 and state 2, respectively. This is the energy that the system has due to its position or configuration.

The second part of the equation, H12 and H21, represents the interaction energy between state 1 and state 2. This interaction energy can arise from various sources, such as electric or magnetic fields, or the presence of other particles in the system.

The term (H12)C2 in the equation specifically refers to the interaction energy between state 1 and state 2, multiplied by the coefficient C2, which represents the probability amplitude for the system to be in state 2. This term takes into account the fact that the interaction energy will have a different contribution to the total energy of the system depending on the probability of the system being in state 2.

In summary, the Hamiltonian matrix equation 8.43 is a representation of the total energy of a quantum system, taking into account both the potential energy and the interaction energy between different states. The term (H12)C2 represents the contribution of the interaction energy to the total energy, weighted by the probability of the system being in state 2.
 

1. What is a Hamiltonian matrix?

The Hamiltonian matrix is a mathematical operator used in quantum mechanics to describe the total energy of a system. It is represented by a square matrix and is an essential component in solving the Schrödinger equation, which determines the time evolution of a quantum system.

2. How is the Hamiltonian matrix used in Feynman III Quantum Mechanics?

In Feynman III Quantum Mechanics, the Hamiltonian matrix is used to calculate the energy levels and transition probabilities of a quantum system. It is also used to describe the interactions between particles and fields, which allows for the prediction of particle behavior.

3. What is the significance of Eq. 8.43 in Feynman III Quantum Mechanics?

Eq. 8.43 in Feynman III Quantum Mechanics is a specific equation that represents the time-independent Schrödinger equation, which can be solved using the Hamiltonian matrix. This equation is essential in understanding the behavior of quantum systems and making predictions about their energy levels and transitions.

4. How does the Hamiltonian matrix relate to Feynman diagrams?

Feynman diagrams are graphical representations of particle interactions in quantum field theory. The Hamiltonian matrix is used to calculate the amplitudes of these interactions, which are then plotted on the Feynman diagram. This allows for a visual representation of the mathematical calculations involved in describing particle behavior.

5. Is the Hamiltonian matrix unique to quantum mechanics?

The concept of a Hamiltonian matrix originated in classical mechanics and was later adapted for use in quantum mechanics. However, it is a fundamental tool in understanding quantum systems and is not used in the same way in classical mechanics. Therefore, while the concept is not unique to quantum mechanics, its application in this field is essential and differs from its use in classical mechanics.

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