Wave packet and uncertainty principle, a mathematical approach

[apalopohapa]

Hello. I started reading this little book by Heisenberg. It starts giving a mathematical relation of width of wave packet and range of wavelengths necessary to mathematically construct it, and then drops in wavalength-momentum relations to give a quick insight on the nature of the uncertainty principle. But I'm just trying to understand the wave packet math part, not even its relation to physical properties of wave-particles.

So, basically, a wave packet (one dimension) is mathematically constructed by the addition of sine waves, all around some wavelength $$\lambda$$. This wave packet has amplitude different to zero within a range in x, this range being denoted as $$\Delta$$x. Now, if $$\lambda$$ is a few times smaller than $$\Delta$$x, there will be a few troughs inside that range, this number "n" being $$\Delta$$x/$$\lambda$$. So far, so good.

But then, just like that, he goes on to say that the wave packet is constructable ONLY if the wavelength range (denoted as $$\Delta$$$$\lambda$$) has a lower limit of at least $$\Delta$$x/(n+1). In other words, the sine component of lowest wavelength has to be at least small enough to fit n+1 troughs within the wave packet width $$\Delta$$x. In this way there is a compromise between the width of the packet and the range of wavelengths of sine components that make it, assuming a main wavelength $$\lambda$$. That's it, no further explanation. How does he know this for sure? What is the proof of this?

This is very important because it is the mathematical origin of the inequality. Then it is simple algebra to arrive to the uncertainty principle, assuming empirically found relations between wavelength and momentum, and interpreting the packet width as uncertainty in position.

 

[mmwave]

Hello, thank you for sharing your thoughts and questions on Heisenberg's book. I can understand your curiosity and desire to understand the mathematical basis behind the uncertainty principle.

Firstly, I want to clarify that the uncertainty principle is a fundamental concept in quantum mechanics, and it is not solely based on the mathematical construction of a wave packet. The uncertainty principle states that it is impossible to precisely know both the position and momentum of a particle at the same time. This principle has been experimentally verified and is a cornerstone of quantum mechanics.

Now, coming to the mathematical construction of a wave packet, it is important to understand that a wave packet is not a physical entity, but rather a mathematical representation of a particle's wave-like behavior. In other words, it is a way to describe the probability of finding a particle at a certain position and momentum.

The mathematical construction of a wave packet involves adding together sine waves of different wavelengths. The narrower the range of wavelengths, the more localized the wave packet will be in space. This is because the more specific the range of wavelengths, the more specific the position and momentum of the particle will be.

Now, let's look at the specific relation mentioned in the book - \Delta\lambda > \Deltax/(n+1). This is known as the uncertainty relation for a wave packet, and it is derived from the mathematical properties of sine waves. Essentially, it means that the range of wavelengths in a wave packet must be larger than the width of the packet divided by the number of troughs within that width plus one. This ensures that the wave packet is constructable and represents a localized particle.

But how do we know this for sure? Well, it is based on the mathematical properties of sine waves and their relationship to position and momentum. The proof of this relation involves complex mathematical calculations and is beyond the scope of this forum. However, it is important to note that this relation is not arbitrary, but rather a fundamental property of waves and their behavior.

I hope this explanation helps you understand the mathematical origin of the uncertainty principle. It is important to remember that the uncertainty principle is not just based on the mathematical construction of a wave packet, but rather a fundamental principle in quantum mechanics.

 

[oswaler]

I can understand your confusion and desire for further explanation. The mathematical approach to understanding the wave packet and uncertainty principle is indeed complex and can be difficult to grasp at first. However, the reason why the wave packet must have a lower limit for its wavelength range is due to the nature of waves and their interference patterns.

When we construct a wave packet, we are essentially combining multiple waves with different wavelengths. However, these waves can only interfere constructively within a certain range of wavelengths. If the wavelength range is too narrow, there will not be enough interference to create a well-defined wave packet. On the other hand, if the wavelength range is too wide, there will be destructive interference and the wave packet will not be well-defined either.

This is why there is a compromise between the width of the packet and the range of wavelengths that make it. The minimum wavelength range required for constructive interference to occur is \Deltax/(n+1), where n is the number of troughs within the packet. This is a fundamental property of waves and can be mathematically proven using wave equations and principles of interference.

From this fundamental understanding of wave packets, we can then derive the uncertainty principle by incorporating the empirically found relations between wavelength and momentum. This is where the mathematical approach connects with the physical properties of wave-particles.

I hope this helps in your understanding of the mathematical approach to the wave packet and uncertainty principle. It is a complex topic, but with further study and exploration, it will become clearer. Keep asking questions and seeking answers, as that is the essence of science.