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In this post, we’ll discuss what solving Schrödinger’s equation brings us by considering the physical meaning of the wave function.
Schrödinger’s equation
$$\left[ -\frac{ħ^2}{2m}∇^2 +V({\bf r}) \right]Ψ({\bf r},t)=iħ\frac{∂}{∂t}Ψ({\bf r},t)$$
We’ll get the following equation if the wave function \(ψ({\bf r})\) does not depend on time \(t\).
\(ħ=h/2π\):Dirac’s constant \(m\):The mass of the particle
\(V\):The potential energy \(E\):The mechanical energy of the particle.
We define Hamiltonian \(\hat{H}\) like below.
$$\hat{H}≡ -\frac{ħ^2}{2m}∇^2 +V({\bf r})$$
Then we find the following equation.
$$\hat{H}ψ=Eψ$$
What is the physical meaning of the wave function \(Ψ({\bf r},t)\)?
Existence probability
The position of particles which are very small like electrons can not be expressed by referring to one point on coordinates such as “exists at the point (a, b, c)” because such particles can be considered as both particles and waves.
Instead, we can find the approximate location of the particles by using the probability that the particles are found within the observed range. This probability is called the existence probability.
What Does Schrödinger’s Equation Bring You?
The wave function \(Ψ(x,t)\) of the Schrödinger equation has a physical meaning for the first time by taking the square of the absolute value \(|Ψ(x,t)|^2 \). This \(|Ψ(x,t)|^2\) is the probability density. The reason why it is called probability “density” is that \(|Ψ(x,t)|^2･dx\) is the existence probability. \(dx\) is the observed range.
In summary, finding the wave function \ (Ψ (x, t) \) by solving Schrödinger equation is finding the approximate location of the particle by finding the existence probability.
Conclusion
･By solving the Schroedinger equation, probability density \(|Ψ({\bf r},t)|^2\) is obtained.