# Quantum mechanics-hamiltonian matrix and stationary states

• mimocs
In summary, the conversation discusses the concept of mutually orthogonal states and the Hamiltonian operator. The Hamiltonian operator is given by H=c[1><2]+c[2><1], where c is a real number. Part (a) of the problem involves calculating the eigenstates and corresponding eigenvalues of H. For part (b), the problem asks how the state evolves over time if the initial state is [1>. The solution involves finding the time dependence of [1(t)> and writing it in terms of [1(0)> and [2(0)>, and solving for the coefficients.
mimocs

## Homework Statement

let [1> and [2> mutually orthogonal states (eigenstates of some Hermitian operator).
the Hamiltonian operator is given by H=c[1><2]+c[2><1], where c is a real number.

(a) calculate the eigenstates and corresponding eigenvalues of H
(b) if the initial state of the system is [1>, how does the state evolve with time?

## Homework Equations

H[ψ>=ih$\frac{∂[ψ>}{∂t}$

## The Attempt at a Solution

for (a), I first earned hamiltonian matrix as below
0 c
c 0

from this, I earned two eigenvalues +c and -c.
and earned $\sqrt{1/2}$( 1 1 ) and $\sqrt{1/2}$( 1 -1 ) as two eigenstates of hamiltonian matrix.

however what I don't know is the answer of (b)
the problem I was given does not have any information of the state [1>. Is there anyway to solve this problem? if I assume [1> as (1 0) (and [2> as (0 1)) this problem can be solved. but I don't know more... can anyone help me?

nice work for 'part a'. I get the same answer. And for part b, it says you need to work out the time dependence. I think there are a couple of slightly different ways you could go about this. The way I prefer to do it is to think about [1(t)>, and write it out in the basis of [1(0)> and [2(0)>. So, you are right that in this basis, [1(t)> starts off as (1 0) But at some time later it will evolve to other values, for example (0 -1). So, the problem is what kind of function of time should these coefficients be? To work that out, you can use your 'relevant equation', and you can also make use of the energy eigenstates if you want, since you have already calculated them. but in the end, I think you should write [1(t)> in terms of [1(0)> and [2(0)>

## 1. What is a Hamiltonian matrix in quantum mechanics?

The Hamiltonian matrix is a mathematical representation of the total energy of a quantum system. It includes the kinetic and potential energies of all particles in the system, and is used in the Schrödinger equation to describe the time evolution of the system.

## 2. How are stationary states related to the Hamiltonian matrix?

Stationary states, also known as eigenstates, are solutions to the Schrödinger equation with a constant energy value. These energy values correspond to the eigenvalues of the Hamiltonian matrix. In other words, the Hamiltonian matrix helps us determine the allowed energy states of a quantum system.

## 3. Can the Hamiltonian matrix be diagonalized?

Yes, the Hamiltonian matrix can be diagonalized, meaning that it can be transformed into a diagonal matrix with the eigenvalues along the diagonal. This is useful for finding the energy levels and corresponding wavefunctions of a quantum system.

## 4. How does the Hamiltonian matrix change for different quantum systems?

The Hamiltonian matrix is specific to each quantum system and depends on the potential energy of the system. For example, the Hamiltonian matrix for a particle in a box will be different from that of a harmonic oscillator. It also changes if there are interactions between particles in the system.

## 5. What role does the Hamiltonian matrix play in quantum mechanics?

The Hamiltonian matrix is a fundamental concept in quantum mechanics, as it helps us understand the energy levels and dynamics of quantum systems. It is used in many calculations and equations, such as the time-dependent and time-independent Schrödinger equations, and is crucial for predicting the behavior of quantum systems.

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