Discussion of the problem of one dimensional harmonic oscillator in quantum mechanics

The Hamiltonian operator

H = Px²/2m + 1/2 mw²X²

Px is the momentum operator

X is the position operator

We introduce two non Hermitian operators

It is known that

Expression of Hamiltonian operator in terms of a and a+

1. "a" and "a†" Operators: In quantum mechanics, the "a" and "a†" operators are related to the annihilation and creation of particles in the context of the harmonic oscillator. These operators help describe the lowering (annihilation) and raising (creation) of energy levels in the oscillator.

2. "ℏ" (h-bar): This symbol represents the reduced Planck constant, which is a fundamental constant in quantum mechanics. It relates the energy and frequency of a quantum system.

3. "w" (omega): In the context of the harmonic oscillator, "w" usually represents the angular frequency, which is related to the oscillation frequency of the oscillator.

4. "H": This likely represents the Hamiltonian operator, which is the quantum mechanical operator corresponding to the total energy of the system.

5. "1/2 ℏw": This term represents half of the energy associated with the angular frequency "w." In the context of the harmonic oscillator, this term is often used in calculations involving energy levels.

   The full equation, therefore, relates the sum of the annihilation and creation operators to an expression involving the energy of the harmonic oscillator. However, the specific form of this equation would depend on the definitions of the operators a and a† and the properties of the quantum harmonic oscillator.

   The Hamiltonian operator for a one-dimensional harmonic oscillator can indeed be expressed in terms of the creation and annihilation operators. The general expression for the Hamiltonian operator in terms of these operators is:

   H = ℏω (a†a + 1/2)

   In this expression, "a" is the annihilation operator, "a†" is the creation operator, ℏ is the reduced Planck constant, and ω is the angular frequency. The term "a†a" represents the number operator, which counts the number of particles in the oscillator's energy levels. The expression "1/2" is an energy offset.

It is the required expression of Hamiltonian operator in terms of a and a†.

Expressing the Hamiltonian operator of a quantum system in terms of the creation (a†) and annihilation (a) operators offers several advantages and insights in the realm of quantum mechanics and quantum field theory:

1. Simplicity in Operator Manipulation: The creation and annihilation operators simplify the mathematical representation of the Hamiltonian, especially in systems with a quantized energy spectrum like the harmonic oscillator. These operators allow for compact and elegant expressions that are easier to manipulate in calculations.

2. Eigenvector Representation: The creation and annihilation operators are directly related to the energy eigenstates of the system. Expressing the Hamiltonian in terms of these operators helps represent energy eigenstates more naturally and allows for straightforward calculations of eigenvalues and eigenstates.

3. Deriving Energy Eigenvalues: The expression of the Hamiltonian in terms of the creation and annihilation operators simplifies the derivation of the energy eigenvalues of the system. This is particularly useful in quantum systems with discrete energy levels, where the energy eigenvalues correspond to observable quantities.

4. Quantum Field Theory: In quantum field theory, creation and annihilation operators are fundamental tools for describing quantized fields. Expressing the Hamiltonian in terms of these operators enables the formulation of quantum field theories and facilitates calculations involving particles and their interactions.

5. Transition Amplitudes and Matrix Elements: The creation and annihilation operators provide a concise way to calculate transition amplitudes and matrix elements between different energy states. This is important for understanding how a quantum system evolves over time and how it transitions between various states.

6. Connection to Experimental Observables: The expression of the Hamiltonian in terms of creation and annihilation operators allows for a direct connection to observable quantities that can be measured in experiments. This facilitates predictions and comparisons between theoretical calculations and experimental results.

7. Manipulating Quantum States: Creation and annihilation operators make it easier to manipulate and analyze quantum states and their properties, such as particle number distributions and statistical properties.

8. Generalization to Many-Body Systems: The use of creation and annihilation operators is crucial when dealing with many-body quantum systems, where multiple particles interact. These operators simplify the description and analysis of complex quantum systems.

In summary, expressing the Hamiltonian operator in terms of the creation and annihilation operators provides a powerful and versatile framework for understanding and calculating properties of quantum systems. It simplifies calculations, provides direct connections to observables, and is essential for both theoretical and experimental studies in quantum mechanics and quantum field theory.

This note is a part of the Physics Repository.