Free particle wave equation

[sniffer]

for a free particle, the wave equation is a superposition of plane waves,

$$\psi(x,0)= \int_{-\infty}^{\infty}g(k)\exp(ikx)dk$$
and
$$g(k)= \int_{-\infty}^{\infty}\psi(0,0)\exp(-ikx)dx$$

one is the Fourier transform of the other. some cases to solve this is when we assume a small delta k, so g(k) behaves like a pulse "delta" function.

is there any more general case we can solve this?

i have been thinking hard if we have definite periodic x, say from 0 up to $2\pi L$, is it solvable?

what would be the (periodic) eigen energy function (if it is)?

 

[vanesch]

for a free particle, the wave equation is a superposition of plane waves,

$$\psi(x,0)= \int_{-\infty}^{\infty}g(k)\exp(ikx)dk$$
and
$$g(k)= \int_{-\infty}^{\infty}\psi(0,0)\exp(-ikx)dx$$

one is the Fourier transform of the other. some cases to solve this is when we assume a small delta k, so g(k) behaves like a pulse "delta" function.

The above pair is, as you say, a Fourier transform pair. Any "nice" function can be written that way, so the above is not a "wave equation" or something, it is a general way of writing a function.
The quantum state of a single scalar particle is described by just such a nice function, called the wave function. At any time, it can be (almost) any function. However, what the wave equation (not written here) gives you, is how this wavefunction AT A CERTAIN TIME t0 will change into the wavefunction at another time t1. This equation will be different according to the situation at hand (free particle, particle in a potential...).

cheers,
Patrick.

 

[quicknote]

The free particle wave equation is a fundamental concept in quantum mechanics and has been extensively studied by scientists. It describes the wave-like behavior of a particle in a free state, meaning it is not subject to any external forces or interactions. The equation is a superposition of plane waves, which represent the different possible states or positions of the particle.

In terms of solving this equation, there are various methods and techniques that have been developed. One approach is through the use of Fourier transforms, as shown in the equations provided. This allows us to express the wave function in terms of its frequency components, making it easier to analyze and solve.

However, there are also other methods and techniques that can be used to solve the free particle wave equation. For example, numerical methods such as finite difference or finite element methods can be used to approximate the solution. In some cases, analytical solutions can also be obtained for specific boundary conditions or assumptions.

Regarding the specific question about solving the equation for a periodic x, there are indeed solutions that exist for this case. These solutions are known as Bloch functions and they represent the eigenfunctions of a periodic potential. The eigen energy function, also known as the energy band structure, can be calculated for a specific periodic potential using the Bloch functions.

In summary, the free particle wave equation is a highly studied and important concept in quantum mechanics. While there are different approaches to solving it, solutions can be obtained through various methods such as Fourier transforms, numerical methods, and Bloch functions for periodic potentials.