That's the main basis for mathematical physics. Why represent force as a vector? Why is time a parameter in non-relativistic physics? Why have a spacetime manifold in relativity?zb23 said:I mean I can't be satisfied with the argument that it gives nice solutions.
Can you give some specfic references to arguments that you find lacking?zb23 said:some arguments for plane wave solutions, for me, lack some structure
You don't have to do it in that old-fashioned way.zb23 said:Why in order to derive the QM momentum operator we use the plane wave solution.
The momentum operator in quantum mechanics is an operator that corresponds to the classical concept of momentum. In the position representation, it is typically denoted as \( \hat{p} \) and is defined as \( \hat{p} = -i\hbar \frac{\partial}{\partial x} \), where \( \hbar \) is the reduced Planck constant and \( x \) is the position variable.
We use the plane wave solution in quantum mechanics because it is an eigenfunction of the momentum operator. Plane waves have the form \( \psi(x) = Ae^{ikx} \), where \( k \) is the wave number. When the momentum operator acts on a plane wave, it yields a simple eigenvalue equation \( \hat{p}\psi(x) = \hbar k \psi(x) \), making it easier to analyze and solve problems involving momentum.
The plane wave solution is directly related to the de Broglie hypothesis, which posits that particles have wave-like properties. According to de Broglie, the wavelength \( \lambda \) of a particle is related to its momentum \( p \) by \( \lambda = \frac{h}{p} \), where \( h \) is Planck's constant. The plane wave \( \psi(x) = Ae^{ikx} \) embodies this wave-particle duality, with the wave number \( k \) related to the momentum by \( p = \hbar k \).
The significance of the plane wave being an eigenfunction of the momentum operator is that it allows for a straightforward interpretation of momentum in quantum mechanics. When a plane wave is used, the momentum operator acting on it yields a definite value (the eigenvalue), which corresponds to the momentum of the particle. This simplifies the analysis of quantum systems and helps in understanding the quantization of momentum.
No, the plane wave solution cannot be used for all quantum mechanical problems. It is ideal for free particles and situations where the potential is constant or negligible. However, for systems with varying potentials, such as particles in a potential well or in the presence of external forces, more complex wavefunctions that satisfy the Schrödinger equation under those specific conditions are required.