# Hamiltonian For The Simple Harmonic Oscillator

• morangta
In summary, the article discusses the "energy surface" of a Hamiltonian for a simple harmonic oscillator with one degree of freedom. It states that the dimensionality of phase space is 2N while the dimensionality of the energy surface is 2N-1. Adjusting x changes the equation for H to H = ω(p*p + x*x), resulting in a 3-dimensional paraboloid when plotted against the x-p plane. Lines of constant H form 2-dimensional circles in the p-x plane. The energy surface for this case would be one-dimensional. The harmonic oscillator equation always determines a hypersphere, making the energy surface one dimension less than the phase space volume.
morangta
I am reading an article on the "energy surface" of a Hamiltonian. For a simple harmonic oscillator, I am assuming this "energy surface" has one (1) degree of freedom. For this case, the article states that the "dimensionality of phase space" = 2N = 2 and "dimensionality of the energy surface" = 2N-1 = 1.

Adjusting x gives H = ω (p + x). That's H = ω(p*p + x*x).

When I plot the energy surface for H, I get a 3-dimensional paraboloid plotted against the x-p plane. Or, lines of constant H are 2-dimensional circles in the p-x plane. What energy surface would be 2N-1 = one (1)-dimensional here? Am not trying to quibble about terminology here. Just want to know if I am missing something.

Hi.
I assume N refers to the number of spatial dimensions, so in the case of a one-dimensional oscillator your phase space is indeed two-dimensional while the energy "surface" is a line (i.e. one-dimensional).
In general, you understand how phase space volume would be 2N-dimensional; now the harmonic oscillator equation always determines a hypersphere (E = x^2 + y^2 +...), so the dimensionality of the hyper-surface is naturally one dimension less than the volume: 2N–1...

hisacro
Ted

## 1. What is the Hamiltonian for the simple harmonic oscillator?

The Hamiltonian for the simple harmonic oscillator is a mathematical expression that represents the total energy of the system. It is given by the equation H = T + V, where T is the kinetic energy and V is the potential energy.

## 2. How is the Hamiltonian derived for the simple harmonic oscillator?

The Hamiltonian for the simple harmonic oscillator can be derived using the principles of classical mechanics and the equations of motion for a harmonic oscillator. It can also be derived using the quantum mechanical approach, where it represents the operator corresponding to the total energy of the system.

## 3. What are the key properties of the Hamiltonian for the simple harmonic oscillator?

The Hamiltonian for the simple harmonic oscillator has several key properties, including: it is a conserved quantity, meaning it remains constant over time; it is quadratic in position and momentum; and it is Hermitian, meaning it is equal to its own conjugate transpose.

## 4. How does the Hamiltonian for the simple harmonic oscillator relate to the Schrödinger equation?

The Hamiltonian for the simple harmonic oscillator is an important component in the Schrödinger equation, which describes the time evolution of a quantum system. The Hamiltonian acts as the operator for the total energy in the equation and helps determine the wave function of the system at different points in time.

## 5. Can the Hamiltonian for the simple harmonic oscillator be extended to more complex systems?

Yes, the Hamiltonian for the simple harmonic oscillator can be extended to more complex systems by including additional terms for other forms of energy, such as rotational or vibrational energy. This allows for the application of the Hamiltonian to a wide range of physical systems, making it a powerful tool in physics and other related fields.

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