What Is the Dimension of the Coefficient b(k) in Quantum Mechanics?

[najima]

Homework Statement

if we have the particle ins free its hamiltonian has a continuous spectrum of eigen enegies and superposition of arbitrary initial state in eigenstates φ_k of H( hamiltonian oprator) becomes ∫_(-∞)^∞▒〖b(k) φ_k dk〗,what is the dimemension of b(k) (lb(k)l^2 is a probability density)? where k is wave nomber.

Homework Equations

quantum mechanics by liboff chapter 5

The Attempt at a Solution

lb(k)l^2 dimensions is 1/k because ∫_(-∞)^∞▒〖b(k)^2dk〗=1

 

[vacuum]

, so b(k) must have dimensions of 1/√k. This means that the dimension of b(k) is 1/√(length^-1), which is equivalent to √length. This dimension is often referred to as the square root of length, and it is commonly used in quantum mechanics to describe the probability amplitude of a particle in a given state. Therefore, the dimension of b(k) is √length.