# Number Operator in Matrix Form

• martyg314
In summary, the conversation is about a problem involving Dirac notation and Hermitian operators for a Spring Break assignment. The first part of the problem involves expressing the number operator in matrix form and showing that it is Hermitian. The second part involves expressing the basis states of the harmonic oscillator problem in column vector form and using the matrix form of the number operator to show the eigenvalue equation. The person is stuck on finding the column vector form of the basis states and is seeking help. They mention that they have already determined the matrix form of the ladder operators, but cannot figure out the column vector form of the basis states. They are looking for guidance on how to arrive at the column vector form and are familiar with the conventional method of
martyg314
Hi-

I have a basic QM problem I am trying to solve. We are just starting on the formalities of Dirac notation and Hermitian operators and were given a proof to do over Spring Break. I am stuck on how to set up the operators and wave equation in matrix and vector form to complete the proof as requested. We have done very basic matrix operations in class (ie $\hat{H}$ =(h g;h g) or a wavefunction in terms of |1> and |0>, but nothing like the following, nor is it covered in the text (Griffiths).

## Homework Statement

Consider the number operator $\hat{N}$ =$\hat{a+}$$\hat{a-}$
for the HO problem.
1) Express the operator in matrix form and show that it is Hermitian.
2) Express the basis states of the HO problem |$\psi$n> in column vector form and use the matrix form of $\hat{N}$ to show that the matrix-mechanics version of the eigenvalue equation ( $\hat{N}$|$\psi$n> = n|$\psi$n> ) works out.

## The Attempt at a Solution

for 1) I determined the matrix form of the ladder operators as the square roots of n, n+1 etc. off the diagonal (ie: {0 0 0; $\sqrt{1}$ 0 0;0 $\sqrt{2}$ 0}). This is easy enough to show as Hermitian.

However I'm stumped as to the column vector form of the HO basis states. I'm sure it's something simple I'm overlooking, but I would appreciate any tips.

Thanks,
M

If you've constructed the matrix form for the ladder operators, then you must already know how to express the basis states as column vectors, because you can't write down a matrix until you know the basis vectors. Can you go into more detail about how you arrived at those matrix forms?

Also, just to check...you are already familiar with the "conventional" way of constructing the basis states for the HO (i.e. using the ladder operators + a vacuum state), correct?

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## 1. What is the Number Operator in Matrix Form?

The Number Operator in Matrix Form is a mathematical representation of the concept of counting or measuring the number of particles in a system. It is commonly used in quantum mechanics to describe the state of a quantum system and predict the outcome of measurements.

## 2. How is the Number Operator represented in a matrix?

The Number Operator is typically represented as a diagonal matrix with the number of rows and columns equal to the number of possible states in the system. The diagonal elements of the matrix correspond to the number of particles in each state, while the off-diagonal elements are all zero.

## 3. What is the significance of the Number Operator in Quantum Mechanics?

The Number Operator is an important tool in quantum mechanics as it allows us to calculate the probabilities of different measurement outcomes and determine the average number of particles in a given state. It also plays a crucial role in understanding the behavior of quantum systems, such as in the study of quantum entanglement.

## 4. How does the Number Operator relate to the Uncertainty Principle?

The Uncertainty Principle states that it is impossible to know both the exact position and momentum of a particle at the same time. The Number Operator is one of the operators used in the mathematical formulation of the Uncertainty Principle, along with the position and momentum operators.

## 5. Can the Number Operator be used in classical mechanics?

No, the Number Operator is a concept that is specific to quantum mechanics and cannot be applied in classical mechanics. In classical mechanics, the number of particles in a system is considered to be a definite and measurable quantity, while in quantum mechanics, it is described probabilistically by the Number Operator.

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