Problem with changing basis in Hilbert space

In summary, the theorem states that a general state of a system can be represented by a unique linear combination of the eigenstates of any Hermitian operator. However, this is not always true, and for some operators it is difficult to prove this.
  • #1
Chain
35
3
The expansion theorem in quantum mechanics states that a general state of a system can be represented by a unique linear combination of the eigenstates of any Hermitian operator.

If that's the case then that would imply we would be able to represent the spin state of a particle in terms of it's energy eigenstates for example however surely it's impossible to do this for a particle in an infinite square well for instance, since the energy eigenstates are independent of spin.
 
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  • #2
I don't see how that implication comes from the statement... You can represent the spin with a unique linear combination of the spin eigenstates, you can represent the energy state as a linear combination of energy eigenstates. Is this not right?
 
  • #3
There are two difficulties.

... a general state of a system can be represented by a unique linear combination of the eigenstates of any Hermitian operator.

This is not necessarily always true. In quantum theory, it can be proven only for certain operators. I think perhaps the Hamiltonian of a harmonic oscillator is a good example in this respect. But I have read somewhere that it is difficult to prove generally that eigenfunctions of every Hermitian operator will form a basis. Often this has to be just assumed.

we would be able to represent the spin state of a particle in terms of it's energy eigenstates for example

No, of course this is not so. The above theorem or assumption applies to single Hilbert space. Spin state and coordinate wave functions belong to different Hilbert spaces.
 
  • #4
Jano L. said:
[...] But I have read somewhere that it is difficult to prove generally that eigenfunctions of every Hermitian operator will form a basis.[...]

Indeed, the nuclear spectral theorem is hard to prove, but nonetheless correct.

Jano L said:
[...] Often this has to be just assumed.[...].

It's always assumed, if you minimize the mathematical rigurosity of your treatment.

Jano L said:
[...] The above theorem or assumption applies to single Hilbert space. Spin state and coordinate wave functions belong to different Hilbert spaces.

The product of the two (i.e. the wavefunction of the system) belongs to the direct product (rigged) Hilbert space.
 
  • #5
Ah okay fair enough guys, I thought it literally meant a wavefunction could be expanded as the eigenstates of any Hermitian operator, so basically the theorem is true only if the observables belong to the same Hilbert space?

Thanks for the replies :)
 

Related to Problem with changing basis in Hilbert space

1. What is a Hilbert space?

A Hilbert space is a mathematical concept that represents an infinite-dimensional vector space. It is used in functional analysis and quantum mechanics to describe the mathematical properties of physical systems.

2. Why is changing basis in Hilbert space important?

Changing basis in Hilbert space allows us to simplify mathematical calculations and better understand the properties of a physical system. It also allows us to manipulate and transform the system in a more intuitive way.

3. What are the challenges associated with changing basis in Hilbert space?

The main challenge is that the mathematical operations involved can be complex and difficult to visualize, especially in higher dimensions. It also requires a thorough understanding of linear algebra and functional analysis.

4. How does changing basis affect the properties of a system in Hilbert space?

Changing basis can affect the properties of a system in different ways, depending on the specific transformation. It can change the values of physical observables, the energy levels of a system, and the probability amplitudes of quantum states.

5. Can changing basis in Hilbert space be applied to real-world problems?

Yes, changing basis in Hilbert space is widely used in many fields, including physics, engineering, and computer science. It has practical applications in data compression, signal processing, and quantum computing, among others.

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