Exploring \psi(x) in Hilbert Space

In summary, the question is asking for which values of \alpha does the function \psi(x)=\frac{1}{x^{\alpha}} belong in a Hilbert space, with the integral condition \int\psi^{2}=A. The question is not completely specified as the domain of \psi(x) is not specified, but it is later clarified that x can be any value. However, it is noted that a wave function with singularity may not be suitable for physics.
  • #1
zhaiyujia
6
0
[SOLVED] Hilbert Space

Homework Statement


For What Values of [tex]\psi(x)=\frac{1}{x^{\alpha}}[/tex] belong in a Hilbert Sapce?

Homework Equations


[tex]\int x^{a}=\frac{1}{a+1} x^{a+1} [/tex]

The Attempt at a Solution


I tried to use the condition that function in Hilbert space should satisfy:
[tex]\int\psi^{2}=A[/tex] but it seems always infinite exist in x=0 or x=infinite
 
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  • #2
Why are you writing the integral of x^a, when you want to examine the integral of 1/x^{2a}?

Also, have you written down the question completely? What is the domain on which [itex]\psi(x)[/itex] is defined?
 
  • #3
one is alpha and another is a. I just write a integral equation, a = 2*alpha. I think it is not the key point. The question is complete. I asked my professor if there is some constrain of x, she said x can be any value. that is to say the integral will from minus infinite to infinite
 
  • #4
zhaiyujia said:
one is alpha and another is a.
Okay, they both looked the same to me.
The question is complete.
It can not be. There's at least a couple of missing words. Here's one way to write a somewhat more complete question:

For what values of [itex]\alpha [/itex] does [itex]\psi(x)=1/{x^{\alpha}}[/itex] belong in a Hilbert space?

I asked my professor if there is some constrain of x, she said x can be any value. that is to say the integral will from minus infinite to infinite
This needs to be specified in the question. You have not completely specified a function unless you describe its domain.
 
  • #5
Thanks, I explained it in the interval of minus infinite to minus zero and zero to infinite. I guess a wave function with singularity is not a good one in physic...
 

1. What is \psi(x)?

In quantum mechanics, \psi(x) is a mathematical function that describes the probability of finding a particle at a specific position x in space.

2. What is Hilbert Space?

Hilbert Space is a mathematical concept used in quantum mechanics to describe the space in which quantum states live. It is a complex vector space with infinite dimensions and allows for the representation of wave functions such as \psi(x).

3. How is \psi(x) explored in Hilbert Space?

In order to explore \psi(x) in Hilbert Space, scientists use various mathematical techniques and tools such as Fourier transforms, operators, and inner products. These allow for the manipulation and analysis of \psi(x) to understand its properties and behavior.

4. What are the applications of exploring \psi(x) in Hilbert Space?

The exploration of \psi(x) in Hilbert Space has many practical applications, such as predicting the behavior of particles in quantum systems, developing new technologies such as quantum computing, and improving our understanding of the fundamental laws of nature.

5. Are there any limitations to exploring \psi(x) in Hilbert Space?

While Hilbert Space provides a powerful mathematical framework for exploring \psi(x), it does have its limitations. For example, it cannot fully explain the phenomenon of quantum entanglement, and some theories suggest that there may be other mathematical frameworks needed to fully understand quantum systems.

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