# QM: Splitting up the Hamiltonian

• Niles
In summary, the conversation discusses a Hamiltonian and a stationary state, and how the Hamiltonian can be split into two equations using the form of the stationary state. The conversation also mentions using the time-independent Schrödinger equation to understand this concept.
Niles

## Homework Statement

Hi all.

I have a Hamiltonian given by:

$$H = H_x + H_y = -\frac{\hbar^2}{2m}(d^2/dx^2 + d^2/dy^2).$$

Now I have a stationary state on the form $\psi(x,y)=f(x)g(y)$. According to my teacher, then the Hamiltonian can be split up, i.e. we have the two equations:

$$H_x f(x) = E_xf(x) \qquad \text{and}\qquad H_y g(y)=E_yg(y).$$

I can't see why this must be true. Inserting in the time-independent Schrödinger-equation doesn't give me these expressions. What am I missing here?

Thanks in advance.

Niles

If you put it in the time-indep Schrodinger equation, and then if you divide both sides by f(x)g(x), and then if you take one of the two terms on the left hand side to the right hand side, you get : a LHS that is a function of x only, and RHS that's function of y only. This can only be true if both sides are equal to some constant. Try to take it from there.

Ahh, yes. I see.

Thanks for that.

## 1. What is the Hamiltonian in quantum mechanics?

The Hamiltonian is an operator that represents the total energy of a quantum system. It is used to calculate the time evolution of the system and is a key component in solving the Schrödinger equation.

## 2. How is the Hamiltonian split up in quantum mechanics?

The Hamiltonian can be split up into different terms, depending on the system being studied. In general, it can be divided into a kinetic energy term, a potential energy term, and any additional terms that account for other interactions or forces in the system.

## 3. Why is splitting up the Hamiltonian important in quantum mechanics?

Splitting up the Hamiltonian allows for a more detailed analysis of the energy levels and dynamics of a quantum system. It also makes it easier to solve the Schrödinger equation and make predictions about the behavior of the system.

## 4. What is the relationship between the Hamiltonian and the energy of a quantum system?

The Hamiltonian is directly related to the energy of a quantum system. The eigenvalues of the Hamiltonian operator represent the possible energy levels of the system, and the corresponding eigenstates represent the possible states of the system at those energy levels.

## 5. How does the splitting of the Hamiltonian affect the behavior of a quantum system?

The splitting of the Hamiltonian can have a significant impact on the behavior of a quantum system. It can affect the energy levels, dynamics, and even the stability of the system. Different splitting schemes can also lead to different predictions and outcomes for the system.

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