Infinite wave train for an electron?

This trick with the dirac delta function is commonly used in quantum mechanics to simplify normalization calculations and is applicable in similar situations.
  • #1
Sacroiliac
13
1
The normalization of the free (say, electron) in quantum mechanics is
achieved by a trick with the dirac delta function. Typically we write the
orthogonality conditions for u1=c*exp(i*k1*x) and u2=c*exp(i*k2*x) as:

int(u1*u2)=delta(k1-k2)

and then out pops the nomalization constant c=1/sqrt(2*pi*hbar). This is
great and all, but what does it mean - we still violate the normalization
condition over all but one length! So what does it mean to 'delta function
normalize'? There are numerous other simialr situations which lead to the
same kind of issue.

Any thoughts?

Thanks
 
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  • #2
The delta function normalization means that the wavefunction you are trying to normalize is orthogonal to all other states at all points but one, in this case k1=k2. This means that the wavefunction can be normalized over the entire range of x, as long as it is orthogonal at the specified point (in this case k1=k2). This allows us to avoid having to normalize over the entire range of x, which would be much more difficult.
 
  • #3
for bringing up this interesting topic! The concept of a delta function normalization can be a bit confusing at first, but it is an important tool in quantum mechanics that allows us to properly describe the behavior of particles such as electrons.

First, let's clarify what we mean by a "wave train" for an electron. In classical physics, a wave train is a series of waves that are all of the same frequency and amplitude, traveling in the same direction. In quantum mechanics, however, particles such as electrons are described by wave functions, which are complex-valued functions that represent the probability amplitude of the particle at different points in space and time. So, instead of a physical wave, a "wave train" for an electron refers to the oscillating behavior of its wave function.

Now, back to the issue of normalization and the use of the delta function. The delta function is a mathematical tool that allows us to represent a very narrow, infinitely tall spike at a specific point in space. In the context of quantum mechanics, we use the delta function to ensure that the wave function of a particle is normalized, meaning that the total probability of finding the particle somewhere in space is equal to 1.

But as you pointed out, this normalization only holds for a single point in space, and not for all points. This may seem counterintuitive, but it is actually a consequence of the probabilistic nature of quantum mechanics. In classical physics, we can determine the exact position and momentum of a particle, but in quantum mechanics, we can only calculate the probability of finding a particle in a certain position or with a certain momentum.

So, the delta function normalization allows us to properly describe the behavior of electrons and other particles in quantum mechanics. It may seem strange at first, but it is a necessary tool for understanding the probabilistic nature of the quantum world. I hope this helps clarify the concept of a delta function normalization for you.
 

1. What is an infinite wave train for an electron?

An infinite wave train for an electron is a theoretical concept in quantum mechanics that describes the behavior of an electron as it moves through space. It is a continuous and unbroken pattern of oscillating waves that represents the probability of finding the electron at any given location.

2. How is an infinite wave train different from a regular wave?

An infinite wave train for an electron is different from a regular wave in two main ways. First, it is a probability wave, meaning it represents the likelihood of finding the electron at a certain location rather than the actual position of the electron. Second, it is infinite in length, meaning it extends infinitely in both directions, whereas regular waves have a finite wavelength.

3. What is the significance of an infinite wave train for an electron?

The concept of an infinite wave train for an electron is significant because it helps us understand the behavior of electrons in quantum mechanics. It allows us to make predictions about the location and behavior of electrons, which is crucial for understanding the properties of matter and developing new technologies.

4. Can an infinite wave train for an electron be observed?

No, an infinite wave train for an electron cannot be observed directly. This is because it is a purely mathematical concept that represents the probability of finding an electron at a certain location. However, scientists can indirectly observe the effects of an infinite wave train through experiments and measurements.

5. How does an infinite wave train for an electron relate to the uncertainty principle?

An infinite wave train for an electron is directly related to the uncertainty principle, which states that it is impossible to know both the exact position and momentum of a particle at the same time. The infinite wave train represents the uncertainty in the position of an electron, as it is spread out over a large area. This uncertainty is a fundamental aspect of quantum mechanics and has important implications for our understanding of the universe.

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