Proving the Heisenberg Uncertainty Relation | Homework Equations Included

[Ohekatos]

Homework Statement

Could someone please have a look at this?
I am to show that from the inequation
$$\langle\left \psi | \hbar^2D^2 | \psi\right\rangle + mk\langle \left\psi | x^2 | \psi\right\rangle\geq\hbar\sqrt{mk}$$
you can get the Heisenberg uncertainty relation
$$\langle\psi | \hbar^2D^2 | \psi\rangle\langle \left\psi | x^2 | \psi\right\rangle\geq\frac{1}{4}\hbar^2$$
for all normalized functions $$\psi \in S(\mathbb{R})$$

Homework Equations

I know that
$$H_0=\frac{\hbar^2}{2m}D^2+\frac{1}{2}kx^2$$
and that
$$H\psi=\hbar\omega\sum_{n=0}^{\infty}(n+1/2)\langle\Omega_n | \psi\rangle\Omega_n$$
and that $$H_0\psi=H\psi$$ for $$\omega=\sqrt{k/m}$$

The Attempt at a Solution

I tried to square on both sides:
$$\langle\psi | \hbar^2D^2 | \psi\rangle^2+m^2k^2\langle \left\psi | x^2 | \psi\right\rangle^2+2mk\langle \left\psi | x^2 | \psi\right\rangle\langle\psi | \hbar^2D^2 | \psi\rangle \geq\hbar^2 mk$$

But that doesn't seem to work right

 

[Jhero]

.

Thank you for sharing your problem with us. It seems like you are trying to use the Cauchy-Schwarz inequality to prove the Heisenberg uncertainty relation. However, there are a few issues with your attempt.

Firstly, when you square both sides of the inequality, you also need to square the inequality sign. This means that your inequality should become:

\langle\psi | \hbar^2D^2 | \psi\rangle^2+m^2k^2\langle \left\psi | x^2 | \psi\right\rangle^2+2mk\langle \left\psi | x^2 | \psi\right\rangle\langle\psi | \hbar^2D^2 | \psi\rangle \geq\hbar^4 mk

Secondly, the Cauchy-Schwarz inequality states that for any two vectors \textbf{a} and \textbf{b} in a vector space, the following inequality holds:

|\langle\textbf{a}|\textbf{b}\rangle|^2 \leq \langle\textbf{a}|\textbf{a}\rangle \langle\textbf{b}|\textbf{b}\rangle

In your case, \textbf{a} and \textbf{b} are not just any vectors, but operators. Therefore, you cannot simply square both sides of the inequality. Instead, you need to use the fact that the operator norm is defined as the square root of the inner product of an operator with itself:

||\textbf{A}|| = \sqrt{\langle \textbf{A}|\textbf{A}\rangle}

Using this, you can rewrite your inequality as:

||\hbar D||^2||\textbf{x}||^2 \geq \frac{1}{4}\hbar^2

Finally, you need to remember that in quantum mechanics, the Heisenberg uncertainty principle is not a statement about a single state, but about the spread of a state in position and momentum space. Therefore, you need to consider the expectation values of the operators in the inequality, rather than just their norm. This means that your final inequality should look like:

\langle\psi | \hbar^2D^2 | \psi\rangle\langle \left\psi | x^2 | \