# QM Variation Method: Show Equations from ci Parameters

• Old Guy
In summary, the variation principle with parameters ci leads to equations where the matrix elements of the Hamiltonian in an orthonormal basis can be represented in position space. The eigenstates of the Hamiltonian can be used to solve this problem.
Old Guy

## Homework Statement

Show that variation principle (parameters ci) leads to equations
$$\sum\limits_{i = 1}^n {\left\langle i \right|H\left| j \right\rangle c_j = Ec_i {\rm{ where }}} \left\langle j \right|H\left| i \right\rangle = \int {d\textbf{r}^3 \chi _j^* \left( \textbf{r} \right)\left( {H\chi _i \left( \textbf{r} \right)} \right)}$$

## Homework Equations

I've got $$\psi \left( \textbf{r} \right) = \sum\limits_{i = 1}^n {c_i \chi _i \left( \textbf{r} \right)}$$
, but

## The Attempt at a Solution

I'm confused about what's being asked, and what the expected result means. Indices as kets and bras? If they're different vectors within the Hilbert space, won't they be orthonormal implying all i not equal j would be zero? I have a nagging feeling I'm either confused by notation or overlooking something basic.

The left hand side is just the matrix elements of the hamiltonian in some orthonormal basis. If they are eigenstates of the Hamiltonian, this matrix will be diagonal.

The expression they give you for the matrix elements is the position space representation

Beyond that, I don't know what you're asking.

Well, thanks for your response. First, let me say that the problem statement as shown was exactly as presented, and we were given no other information. The only thing I can think of is that the indices in the bras and kets are really intended to be $$\chi_{i}$$ and $$\chi_{j}$$, but this seems to be trivial, because in that case wouldn't $$\left\langle j \right|H\left| i \right\rangle = \int {d\textbf{r}^3 \chi _j^* \left( \textbf{r} \right)\left( {H\chi _i \left( \textbf{r} \right)} \right)}$$
imply $$\left\langle i \right|H\left| j \right\rangle = \int {d\textbf{r}^3 \chi _i^* \left( \textbf{r} \right)\left( {H\chi _j \left( \textbf{r} \right)} \right)}$$
? And if that's the case, doesn't that simply mean that the $$\chi$$'s are simply the eigenstates of the Hamiltonian?

## 1. What is the QM Variation Method?

The QM Variation Method is a mathematical technique used in quantum mechanics to approximate the ground state energy of a system. It involves varying a set of parameters, known as ci parameters, to find the minimum energy of the system.

## 2. How does the QM Variation Method work?

The QM Variation Method works by taking a trial wavefunction, which is a mathematical representation of the system, and varying the ci parameters to minimize the energy of the system. This is done through a series of mathematical equations and calculations.

## 3. What are ci parameters?

Ci parameters are a set of variables that represent the coefficients in a linear combination of basis functions. They are used in the QM Variation Method to adjust the trial wavefunction and minimize the energy of the system.

## 4. What are basis functions?

Basis functions are mathematical functions that are used to represent the wavefunction of a system. They are typically chosen to be orthogonal and can include functions such as polynomials, exponentials, and trigonometric functions.

## 5. How accurate is the QM Variation Method?

The accuracy of the QM Variation Method depends on the choice of basis functions and the number of ci parameters used. With a large number of parameters and a suitable choice of functions, the method can provide a very accurate approximation of the ground state energy of a system.

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