Sage-Tips

Do square integrable functions vanish at infinity?

For the probability interpretation of wave functions to work, the latter have to be square integrable and therefore, they vanish at infinity.

How do we interpret the physical meaning of the square of the wave function?

But the wave function itself has no physical interpretation. However, the square of the absolute value of the wave function has a physical interpretation. We interpret |ψ(x,t)|2 as a probability density, a probability per unit length of finding the particle at a time t at position x.

What is the significance of ψ and ψ2?

ψ is a wave function and refers to the amplitude of electron wave i.e. probability amplitude. It has got no physical significance. The wave function ψ may be positive, negative or imaginary. [ψ]2 is known as probability density and determines the probability of finding an electron at a point within the atom.

Why must a wave function be finite?

Finite. 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.

When a function is Lebesgue integrable?

If f, g are functions such that f = g almost everywhere, then f is Lebesgue integrable if and only if g is Lebesgue integrable, and the integrals of f and g are the same if they exist.

What is a square integrable random variable?

A random variable X is called “integrable” if E|X| < ∞ or, equivalently, if X ∈ L1; it is called “square integrable” if E|X|2 < ∞ or, equivalently, if X ∈ L2. Integrable random variables have well-defined finite means; square-integrable random variables have, in addition, finite variance.

Why does the wavefunction go to zero at infinity?

In order to avoid infinite probabilities, the wave function must be finite everywhere. In order to avoid multiple values of the probability, the wave function must be single valued. In order to normalize the wave functions, they must approach zero as x approaches infinity.

What is the physical interpretation of the wave function how a free particle wave function signifies a particle in space and momentum?

But the wave function itself has no physical interpretation. It is not measurable. However, the square of the absolute value of the wave function has a physical interpretation. We interpret |ψ(x,t)|2 as a probability density, a probability per unit length of finding the particle at a time t at position x.

What is the connection between energy and square-integrable functions?

The connection with square-integrable functions was already firmly established. In hindsight we can see good reasons that square-integrable functions are connected with energy, and other Physics concepts, but the abstraction seems to have been dictated more out of solving equations using ‘orthogonality’ conditions.

What is the definition of square integrability?

In mathematics, a square-integrable function, also called a quadratically integrable function or function, is a real – or complex -valued measurable function for which the integral of the square of the absolute value is finite. Thus, square-integrability on the real line is defined as follows. One may…

What is the history of square-integrable functions?

But square-integrable functions gained interest in the early 19th century, and especially after the early 19th century work of Fourier. It took some time to see a general Cauchy-Schwarz inequality, and to begin to see a connection with geometry, eventually leading to inner-product space abstraction for the space of square-integrable functions.

Why must the wave function go to 0 at ∞?

A consequence of this is that the wave function must go to 0 at ± ∞. The wave function must be continuous everywhere. If this was not true then the momentum of the system, which is found from the first order derivative in space would go to ∞ which is not a physically realizeable system.