# Quantum wavepackets expanding in free space

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• fog37
In summary, the concept of a quantum free particle is usually introduced as a plane wave, which is not physically realizable but serves as a good approximation in many important problems. However, a more realistic representation is a wavepacket, which spreads in free space due to the nature of the dispersion relation. This behavior is similar to a free gas expanding in space, and is a consequence of the lack of forces acting on the particle.
fog37
Hello,

A quantum free particle (no forces acting on it) is usually introduced as a plane wave. A plane wave has a very specific momentum and energy but the function is not normalizable and the particle has a change to be everywhere in space. Mathematically, this plane wave cannot be represented by a real-valued sine wave like sin(kx-wt) which does not solve SE. The plane must be expressed as a complex wave e^i(kx-wt).

Another more realistic way to represent a free particle is as a wavepacket (summation of plane waves). Why bother with the plane wave scenario at all if it represents such an unrealistic case and a pure idealization?

Interestingly, a wavepacket made of light and traveling in a vacuum does not spread (no dispersion): the composing plane waves all travel at the same phase speed, hence the packet does not change shape and continues traveling at the group velocity. However, a quantum wavepacket, like a free electron, spreads in free space, i.e. its probability density spreads, as it travels at its group speed. I can see why that happens mathematically from the nature of the dispersion relation. But what is the overarching concept behind this natural expansion of the probability density? The quantum particle seems to behave like a free gas that continues to expands in free space. What are the consequences behind this? Most problems we deal with seem to involve forces (potentials) that influence them.
So a single electron, or proton or neutron, etc. would spread in free space while a photon does (which I infer from a light pulse not spreading).

Thanks!

fog37 said:
Another more realistic way to represent a free particle is as a wavepacket (summation of plane waves). Why bother with the plane wave scenario at all if it represents such an unrealistic case and a pure idealization?
There are two reasons. First, you can't claim that a superposition of plane waves is a solution to Schrodinger's equation until you've satisfied yourself that the plane waves are themselves a solution of Schrodinger's equation. Second, although it is not physically realizable the plane wave is a very good approximation to a particle of known momentum in many scattering/reflection/transmission problems - and these are some of the most important problems we encounter.

## 1. What is a quantum wavepacket?

A quantum wavepacket is a mathematical representation of a quantum particle that describes its position and momentum in space and time. It is a time-dependent superposition of multiple quantum states.

## 2. How do quantum wavepackets expand in free space?

Quantum wavepackets expand in free space due to the uncertainty principle, which states that the more precisely we know a particle's position, the less precisely we can know its momentum. As a result, the wavepacket spreads out over time as it evolves according to the Schrodinger equation.

## 3. What factors affect the expansion of a quantum wavepacket in free space?

The expansion of a quantum wavepacket in free space is affected by the initial size and shape of the wavepacket, the particle's mass and energy, and the potential energy landscape it is moving through. Additionally, interactions with other particles can also impact the expansion.

## 4. Can quantum wavepackets be controlled or manipulated?

Yes, quantum wavepackets can be controlled and manipulated through various techniques such as shaping the potential energy landscape, using external fields, and applying pulses of light. These methods can be used to steer and shape the wavepacket's expansion for specific applications.

## 5. What practical applications do quantum wavepackets have?

Quantum wavepackets have many applications in fields such as quantum computing, quantum information processing, and quantum metrology. They are also used in studying ultrafast chemical reactions and understanding the behavior of particles at the nanoscale. Additionally, controlling and manipulating wavepackets can help in developing new technologies and devices.

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