Heisenberg Uncertainty Principle question

• jap33
In summary, the given 2 unnormalized wave functions, Y1(x)=e^i(x/m) and Y2(x)=1/2*[e^2i(x/m) + e^3i(x/m) + e^-2i(x/m) + e^-3i(x/m)], suggest that the position of the particles can be measured to be more localized in space by evaluating the uncertainty in position using the equations A = \int_{-L/2}^{L/2}{|\psi(x)|^{2} \, dx}, A \, \langle x \rangle = \int_{-L/2}^{L/2}{x \, |\psi(x)|^{2} \, dx},
jap33
given 2 unnormalized wave functions:

Y1(x)=e^i(x/m)

Y2(x)=1/2*[e^2i(x/m) + e^3i(x/m) + e^-2i(x/m) + e^-3i(x/m)]

if the positions of the particles were measured, which would be found to be more localized in space? (that is, which has a position known more precisely?)

to my understanding, i understand the principle if you know position specifically, then you know nothing about the momentum, etc.

Hint: Take the x-axis to be of finite extent ( $-L/2 \le x \le L/2$). this will make all your integrals converge. Then evaluate:

$$A = \int_{-L/2}^{L/2}{|\psi(x)|^{2} \, dx}$$

$$A \, \langle x \rangle = \int_{-L/2}^{L/2}{x \, |\psi(x)|^{2} \, dx}$$

$$A \, \langle x^{2} \rangle = \int_{-L/2}^{L/2}{x^{2} \, |\psi(x)|^{2} \, dx}$$

Then, once you have calcualated $\langle x \rangle$ and $\langle x^{2} \rangle$, you can evaluate the uncertainty in the position by:

$$\Delta x = \sqrt{\langle (\Delta x)^{2} \rangle} = \sqrt{\langle x^{2} \rangle - \langle x \rangle^{2}}$$

What happens to this number if you set $L \rightarrow \infty$?

What is the Heisenberg Uncertainty Principle?

The Heisenberg Uncertainty Principle is a fundamental concept in quantum mechanics that states that it is impossible to simultaneously know the exact position and momentum of a particle. This means that the more accurately one of these properties is measured, the less accurately the other can be known.

Who discovered the Heisenberg Uncertainty Principle?

The principle was proposed by German physicist Werner Heisenberg in 1927.

What are the implications of the Heisenberg Uncertainty Principle?

The Heisenberg Uncertainty Principle has significant implications for our understanding of the behavior of particles at the subatomic level. It shows that there are inherent limitations in our ability to measure and predict the behavior of particles, and that there is a level of uncertainty and randomness in the universe.

Can the Heisenberg Uncertainty Principle be violated?

No, the Heisenberg Uncertainty Principle is a fundamental principle of quantum mechanics and has been repeatedly confirmed by experiments. It is a fundamental limitation of our ability to understand and predict the behavior of particles.

How does the Heisenberg Uncertainty Principle relate to other principles in physics?

The Heisenberg Uncertainty Principle is closely related to other fundamental principles in physics, such as the wave-particle duality and the principle of complementarity. It also has implications for concepts such as entanglement and the observer effect.

Replies
7
Views
2K
Replies
2
Views
1K
Replies
23
Views
4K
Replies
4
Views
3K
Replies
3
Views
2K
Replies
2
Views
1K
Replies
13
Views
1K
Replies
5
Views
3K
Replies
4
Views
1K
Replies
3
Views
2K