# Infinite quantum wave function

• lark
In summary, the conversation discusses the possibility of a quantum wave function being infinite at a point and whether this is physically possible. The concept of a Dirac delta function, which represents a particle with a precise position, is brought up as an example. The idea of a wave function having a slope that is a continuous function and the concept of idealized limiting cases are also mentioned. The conversation then moves on to discussing the Fourier transform of a plane wave and the meaning of the derivative of a delta function. The idea of a wave function representing probability amplitudes and the unphysical nature of a wave function with infinite expected energy is also discussed. The conversation ends with mention of delta function normalization and the unphysical nature of a plane wave in momentum
lark
Can a quantum wave function be infinite at a point? For example you could have a radially symmetric wavefunction that's infinite at the center, yet the integrated probability is 1. Is this unphysical somehow?
Laura

You are describing the Dirac delta function, the wave function of a particle with a precise position.

A proper wave function must have a slope that is a continuous function. I don't think this is possible at a point where the function goes to infinity. However, such a function might appear as an idealized limiting case of a sharply "peaked" wave function. For example:

clem said:
the Dirac delta function, the wave function of a particle with a precise position.

clem said:
You are describing the Dirac delta function, the wave function of a particle with a precise position.

I was thinking about *something* like probability density = exp(-r)/r. Something that goes infinite at the center yet has finite integral. Perfectly well-behaved except at one point. The Dirac delta function wouldn't ever appear in reality although it might as an intermediate step in one's calculations.
Laura

lark said:
I was thinking about *something* like probability density = exp(-r)/r. Something that goes infinite at the center yet has finite integral. Perfectly well-behaved except at one point. The Dirac delta function wouldn't ever appear in reality although it might as an intermediate step in one's calculations.
Laura

What is the Fourier transform of a plane wave? This will be the wavefunction in momentum space.

Zz.

now we are talking about dirac deltas, the wavefunction

$$D^{n} \delta (x-a)$$ (derivative of delta function) has a meaning ?

from Fourier analysis, we could consider the wave function above the Fourier transform of $$x^{n}$$

If the wave function represents probability amplitudes by definition, how could it be greater than 1 at any point?

because its a probability density, you must integrate it to get the probability the particle is in a given range.this situation is unphysical, it corresponds to a wavefunction with infinite expected energy.

to see this write the delta in momentum space (neglecting the various constants its e^ipx')

now use the momentum space hamiltonian for a free particle ((p^2)/2m)) and take the expectation value.

you would get an infinite result, however this is not the only reason why it is unphysical, what the delta wavefunction means in position space is a plane wave in momentum space, this plane wave in momentum space would not be normalizable and thus unphysical.

Have you heard of delta function normalization?

## 1. What is the "infinite quantum wave function"?

The infinite quantum wave function is a mathematical description of the quantum state of a system, which includes all of its possible states and their probabilities. It is an essential element of quantum mechanics and is used to make predictions about the behavior of particles at the quantum level.

## 2. How is the infinite quantum wave function different from a traditional wave function?

The infinite quantum wave function differs from a traditional wave function in that it includes all possible states of a system, whereas a traditional wave function only describes a single state. This allows the infinite quantum wave function to account for superposition and other quantum phenomena that cannot be explained by traditional wave functions.

## 3. Can the infinite quantum wave function be observed or measured?

No, the infinite quantum wave function is a mathematical construct and cannot be directly observed or measured. However, it can be used to make predictions about the behavior of quantum systems, and these predictions have been confirmed through experiments.

## 4. How is the infinite quantum wave function related to the uncertainty principle?

The infinite quantum wave function is related to the uncertainty principle in that it describes the uncertainty in the position and momentum of a particle at the quantum level. The more accurately one of these properties is known, the less accurately the other can be known, as described by the Heisenberg uncertainty principle.

## 5. What is the significance of the infinite quantum wave function in quantum computing?

The infinite quantum wave function is a crucial aspect of quantum computing as it allows for the manipulation and control of quantum states, which is necessary for performing operations and calculations. Without the infinite quantum wave function, quantum computers would not be able to function as they do.

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