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Heisenberg Uncertainty Principle

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Jetzt kostenlos anmeldenLet's say you have a professional camera, and you want to take pictures of a tennis ball moving. If you began by taking pictures of the ball with a fast shutter speed, you could capture the ball without blur, allowing you to see its position. However, you would be unable to determine the ball's velocity. Conversely, if you try to capture the ball's velocity instead of its position, your photos will appear blurry, and you won't be able to tell the ball's position.

This is a simple analogy to** Heisenberg's uncertainty principle**, which states that we cannot know an object's position and velocity at the same time!

- This article is about the
**Heisenberg unicertainly principle**. - First, we will look at an overview of some
**important concepts**for understanding the Heisenberg uncertainty principle. - Then, we will look at the
**definition of the Heisenberg****uncertainty principle**, and its**formula**. - After, we will briefly look at the
**derivation**of the Heisenberg uncertainty principle formula. - Lastly, we will discuss its
**importance**and**solve a simple problem**.

Before diving into the Heisenberg uncertainty principle, we need to review what **energy leve****ls** are. We know that the structure of an atom consists of neutrons and protons in the atom's nucleus, and electrons "orbiting" the nucleus. These electron orbits are called **energy levels****, **and the electrons in the outermost level are known as** valence electrons**.

An **electron's possible location** around a nucleus can be determined by its **energy level**.

For example, the Bohr model of the element calcium (Ca) shows 20 electrons arranged in four energy levels.

In the process of moving from a higher to a lower energy level (closer to the nucleus), an electron releases energy, emitting a **photon**.

But, how exactly does an electron behave? Does it behave like a wave or like quantum particles of energy? Well, it depends on which scientist you ask!

According to Louis de Broglie, matter possesses both particle and wave properties. de Broglie suggested that objects (such as electrons) can behave as though it moves in a wave. Louis de Broglie came up with the **de Broglie wavelength** formula to calculate the wavelength of matter. This formula relates the wavelength of a moving particle like an electron to its mass and velocity.

$$ \lambda = \frac{h}{m\times v}$$

Where:

- \( \lambda \) is equal to wavelength (in meters).
- \( h \) is equal to planck's constant \( 6.626\times 10^{-34}\text{ J}\cdot \text{s} \).
- \( m \) is equal to the mass of object (in kg).
- \( v \) is equal to the velocity of the object (in m/s).

Later, in 1927, a German physicist called Werner Heisenberg proposed that it is not possible to know where exactly an electron is located when it behaves like a wave and what its velocity is simultaneously because by trying to take any measurements, we would be disturbing it in some way. So, he developed the **Heisenberg uncertainty principle. **

Let's look at the definition of the Heisenberg uncertainty principle.

The **Heisenberg uncertainty principle** states that it is impossible to know the momentum \((mv)\) and position (\(\text{x})\) of an object (for example, an electron) simultaneously and with precision.

- The object's
*velocity*is related to its wave nature, whereas the*position*of the object is related to its particle nature.

In simpler terms, Heisenberg suggested that objects (from electrons to a tennis balls) can be seen as particles or waves, but not both simultaneously!

The Heisenberg uncertainty principle formula is as follows:

$$ \Delta\text{x}\cdot \Delta\text{p}\geqslant \frac{h}{4\pi} $$

Where,

- \( \Delta\text{x} \) = uncertainty in the particle's position (in m).
- \( \Delta\text{p} \) = uncertainty in momentum (in \( kg\cdot \frac{m}{s} \)).
- \( h\) = Planck's constant \( (6.626\times 10^{-34}\text{ J}\cdot \text{s}) \)

The equation to calculate the uncertainty in momentum, \( \Delta {\text {p}} \) is: \( \Delta\text{p = }m\text{ }\times \text{ }\Delta v \). Here, \(m\) is the mass in kg and \( \Delta v \) is the uncertainty in velocity (in m/s).

According to the **Heisenberg uncertainty formula**, the uncertainty in position times the uncertainty in momentum is greater than/equal to Planck's constant \((h)\) divided by four \( \pi \).

According to this formula, the more accurately we know an object's position, the less accurately we can know its momentum (and vice versa). When talking about electrons, the uncertainty principle says that we cannot know the exact motion of an electron as it moves around the nucleus.

Now that we know that the Heisenberg uncertainty principle is, let's look at how it was derived. The best way to explain Heisenberg uncertainty principle is by looking at a** single slit diffraction experiment** using an electron beam. In this experiment, electrons are allowed to pass through a slit, and because of the wave nature of these electrons, they spread out and create a diffraction pattern on a projected screen.

However, we can't know for sure where an electron will hit or land after it passed through the slit, only that will you be somewhere within the slit. This is where** uncertainty in position** \( (\Delta\text{x}) \) comes in. In this case, the uncertainty in position \( (\Delta\text{x}) \) = width of the slit, \( a \).

Similarly, for the electron waves to spread out and produce a single slit diffraction pattern, electrons must have a horizontal velocity (\(v\)), and this velocity differs for each electron. Now, if we try to focus on finding the electron's exact position, then we create an uncertainty in their velocity (\( \Delta v\)).

In short, when the slit gets narrower, the uncertainty in position, \( \Delta\text{x} \), of the electron is decreased, and the spot being projected on the screen starts to spread out. By decreasing \( \Delta\text{x} \), we are increasing the uncertainty in momentum (\( \Delta\text{p} \)).

The simple derivation of the Heisenberg uncertainty principle is shown below. Here, scientists started with the equation for kinetic energy, where \(m\) is the mass and \( v\) is the velocity, and then by using dimensional analysis, they found out that energy (E) multiplied by time (t) is equal to momentum (p) times position (x).

Then, they noticed that \( \text{E }\times \text{t} = ℏ\) (Planck's constant), and since kinetic energy (E) has a 1/2 factor, scientists divided \(ℏ \) by 2, arriving to the uncertainty principle for momentum (p) and position (x):

\( \Delta\text{p }\Delta \text{x }\geqslant \frac{ ℏ}{2} \).

\(\frac{ ℏ}{2} \) is the same as \( \frac{h}{4 \pi}\), making the heinsenberg uncertainty principle formula

\( \Delta\text{x}\cdot \Delta\text{p}\geqslant \frac{h}{4\pi} \).

There is another formula for Heisenberg uncertainty principle for simultaneous measurements of energy and time. In this formula, \( \Delta\text{E}\cdot \Delta\text{t}\geqslant \frac{h}{4\pi} \), where, ΔE is the uncertainty in energy, and Δt is the uncertainty in time.

The importance of the Heisenberg uncertainty principle lies in that, together with the wave/particle duality, it helps indicate how different objects behave at microscopic levels.

Let's finish off by looking at an example of a problem involving Heisenberg uncertainty principle.

**Suppose that you measured an electron's position to an accuracy of \(1.5\times 10^{-11} \) meters. Calculate the electron's uncertainty in velocity (\(\Delta v\)).**

This question gives us uncertainty in position (\(\Delta \text{x}\)), which is \(1.5\times 10^{-11} \) meters, and asks us to find (\(\Delta v\)). Now, to find uncertainty in velocity, we first need to solve for uncertainty in momentum (\(\Delta \text{p}\)) and then use the formula to calculate \(\Delta v\).

**Step 1 - Solve for uncertainty in momentum ***(\(\Delta \text{p}\))***.**

$$ \Delta\text{p}= \frac{h}{4\pi \cdot \Delta\text{x} } $$

$$ \Delta\text{p}= \frac{6.626\times 10^{-34} \text{J}\cdot \text{s}}{4\pi (1.5\times 10^{-11}\text{ m)} } $$

$$ \Delta\text{p}= 3.52\times 10^{-24}\text{ kg}\cdot\frac{\text{m}}{\text{s}} $$

*Step 2- Solve for ***uncertainty in velocity (\(\Delta v***\)) using*** \(\Delta \text{p}\).**

Remember that, the mass of an electron is \( 9.11\times 10^{-31} \text{ kg} \)

$$ \Delta\text{p}= m\times \Delta v $$

$$ \Delta v =\frac{ \Delta\text{p}}{m} $$

$$ \Delta v=\frac{3.52\times 10^{-24} \text{ kg}\cdot \text{m/s}}{9.11\times 10^{-31} \text{ kg}} = 3.86\times 10^{6}\text{ m/s} $$

Now, I hope that you were able to understand Heisenberg uncertainty principle a bit better!

- The
**Heisenberg uncertainty principle**states that it is impossible to know the momentum \((mv)\) and position (\(\text{x})\) of an object (for example, an electron) simultaneously and with precision. - According to the Heisenberg uncertainty principle, the more accurately we know an object's position, the less accurately we can know its momentum (and vice versa).
- According to the
**Heisenberg uncertainty formula**, the uncertainty in position times the uncertainty in momentum is greater than/equal to Planck's constant divided by four \( \pi \). - The general formula for the Heisenberg uncertainty principle is: \( \Delta\text{x}\cdot \Delta\text{p}\geqslant \frac{h}{4\pi} \).

- Jackson, G. (2016, June 23). GM Jackson Physics and Mathematics: A Simple Way to Derive the Heisenberg Uncertainty Principle. GM Jackson Physics and Mathematics. http://gmjacksonphysics.blogspot.com/2016/06/here-is-simple-way-to-derive-heisenberg.html
- Openstax. (2012b). College Physics. Openstax College.
- Randall Dewey Knight, Jones, B., & Field, S. (2019). College physics : a strategic approach. Pearson.
- Swanson, J. (2021). Everything you need to ace chemistry in one big fat notebook. Workman.

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