Table of Contents
Why does the derivative of the wave function have to be continuous?
We require the first derivative ψ (x) to be continuous everywhere except at points where ψ(x) itself is not differentiable; at these points, ψ (x) is allowed to have a finite discontinuity (that is, a step up or down). This happens with the delta function potential. Thus ψ(x) must be continuous.
Why is wave function single valued?
The wave function must be single valued. This means that for any given values of x and t , Ψ(x,t) must have a unique value. This is a way of guaranteeing that there is only a single value for the probability of the system being in a given state.
Which wave has infinite number of continuous derivatives?
A triangle is continuous, but its first derivative is a square wave, which is not continuous. A triangle wave therefore has a infinite series of harmonics.
What is the second derivative of a wave function?
The second derivative of a function is called its curvature. Paying attention to a wave function’s curvature and the way it is controlled by the relative sign of the total energy and the potential energy at a given point will help you understand why a particular wave function looks as it does.
How did Schrodinger come up with the Schrodinger equation?
In his 1924 thesis, de Broglie had proposed a theory of wave mechanics. This sparked Schrödinger’s interest in explaining that an electron in an atom would move as a wave. The following year, he wrote a revolutionary paper that highlighted what would be known as the Schrödinger wave equation.
How to rewrite the Schrodinger equation to avoid the time derivative?
One can rewrite the Schrödinger equation in a way that avoids the time derivative by considering states of definite energy. For such states, E^ψ=Eψ\\hat{E} \\psi = E\\psiE^ψ=Eψ by definition. The Schrodinger equation then becomes H^ψ=Eψ\\hat{H} \\psi = E\\psiH^ψ=Eψ, where EEE is now a constant and not an operator.
What is the time-dependent Schrodinger equation?
The time-dependent Schrodinger equation is the version from the previous section, and it describes the evolution of the wave function for a particle in time and space. A simple case to consider is a free particle because the potential energy V = 0, and the solution takes the form of a plane wave. These solutions have the form:
What is the Schrodinger equation in quantum mechanics?
Schrödinger Equation The Schrödinger equation is a differential equation that governs the behavior of wavefunctions in quantum mechanics. The term “Schrödinger equation” actually refers to two separate equations, often called the time-dependent and time-independent Schrödinger equations.
What is the nonrelativistic Schrödinger equation?
The nonrelativistic Schrödinger equation is a type of partial differential equation called a wave equation. Therefore, it is often said particles can exhibit behavior usually attributed to waves.