# Free particle in quantum mechanics, Dirac formalism

In summary, Shankar wants to find the solution for a free particle in Quantum Mechanics. He states that any eigenstate of the momentum operator P is also an eigenstate of the momentum operator P^{2}. He then says that ''feeding the trial solution \left|p\right\rangle '' into the time independent Schrodinger equation, he gets:\frac{P^{2}}{2m}\left|p\right\rangle = E\left|p\right\rangleBut I don't get it. Why did he change from the \left|E\right\rangle eigenbasis to the \left|p\right\rangle eigen
The problem is very easy, maybe just something about eigenvectors that I'm missing. Go to the first two pages of the 5th chapter of ''Principles of Quantum Mechanics'', by Shankar, 2nd edition.

## Homework Statement

Shankar wants to find the solution for a free particle in Quantum Mechanics. I've got a problem with what he calls ''trial solution'', and the change from an eigenvector to another.

Hamilton operator: $H=\frac{P^{2}}{2m}$ for a free particle

$P$, the momentum operator, $m$ the particle's mass

State vector (ket): $\left|\psi\right\rangle$

## Homework Equations

The Schroedinger equation:

$i\hbar\\left|\dot{\psi}\right\rangle = H\left|\psi\right\rangle = \frac{P^{2}}{2m}\left|\psi\right\rangle$

The time independent Schrodinger equation:

$H\left|E\right\rangle = \frac{P^{2}}{2m}\left|E\right\rangle = E\left|E\right\rangle$

## The Attempt at a Solution

Shankar then says that ''any eigenstate of $P$ is also an eigenstate of $P^{2}$'', which is very clear. Then he says: ''feeding the trial solution $\left|p\right\rangle$ '' into the time independent Schrodinger eq., he gets:

$\frac{P^{2}}{2m}\left|p\right\rangle = E\left|p\right\rangle$

But I don't get it. Why did he change from the $\left|E\right\rangle$ eigenbasis to the $\left|p\right\rangle$ eigenbasis, and still maintained the eigenvalue E? Is there atheorem of eigenbasis I'm missing? I mean, $\left|p\right\rangle$ is an eigenvector of $P$, by definition, and then, of $P^{2}$. Then, from the time independent Schr. eq. we see that $\left|E\right\rangle$ is an eigenvector of $P^{2}$ too. But that doesn't mean it's the same eigenbasis, does it? Or what does ''trial'' mean in this context (forgive me, my mother language is spanish).

After that ''change of basis'' from $\left|E\right\rangle$ to $\left|p\right\rangle$, everything is fine. So please, explain it to me.

Please, forgive me, I know this is a very stupid question, but I just don't get it.

Thankyou very much for your patience.

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The ket $\vert p \rangle$ is an eigenstate of both the momentum operator $\hat{p}$ and the Hamiltonian $\hat{H}$. If you apply Hamiltonian to it, you will get the corresponding eigenvalue of the Hamiltonian, namely E.

The $\vert E \rangle$ and $\vert p \rangle$ bases aren't the same. While a state $\vert p \rangle$ is an eigenstate of $\hat{H}$, a state $\vert E \rangle$ isn't necessarily an eigenstate of $\hat{p}$. For example, you can show that a state with the wave function $\psi(x) = e^{ikx} + e^{-ikx}$ is an eigenstate of $\hat{H}$ with eigenvalue $E=\hbar^2k^2/2m$ but it's not an eigenstate of $\hat{p}$.

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Dear Lord, how could I be so blind??! That's crystal clear! Thank you so much, vela, for taking the time for answering such a silly question! THANK YOU!

## 1. What is a free particle in quantum mechanics?

A free particle in quantum mechanics refers to a particle that is not subject to any external forces or interactions. It is described by a wave function that evolves over time according to the Schrödinger equation.

## 2. What is the Dirac formalism?

The Dirac formalism is a mathematical framework for describing quantum mechanics using operators and state vectors. It was developed by physicist Paul Dirac and is based on the principles of linear algebra.

## 3. How is a free particle described in the Dirac formalism?

In the Dirac formalism, a free particle is described by a ket vector in a Hilbert space. The wave function of the particle is represented by the ket vector and its momentum is represented by the corresponding bra vector. The time evolution of the particle is described by the Schrödinger equation.

## 4. What is the significance of the Dirac equation in the study of free particles?

The Dirac equation is a relativistic version of the Schrödinger equation and is used to describe the behavior of free particles with spin, such as electrons. It takes into account the effects of special relativity and provides a more accurate description of the particles' behavior.

## 5. How does the concept of superposition apply to free particles in quantum mechanics?

In quantum mechanics, a free particle can exist in a state of superposition, meaning it can have multiple possible states or positions at the same time. This is described by the wave function, which contains information about all the possible states of the particle. When the particle is measured, the wave function collapses and the particle is found in one specific state.

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