Rough Description of State Space

In summary, quantum mechanics involves a state space with an infinite number of points and dimensions, similar to an ordinary vector space. The space has many bases, often chosen to be orthonormal for practical reasons. The inner product is defined to ensure the orthogonality of bases.
  • #1
chessforce
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I have done only a small bit of reading/studying of quantum mechanics. So, from what I have gathered thus far, I have the following rough semi-graphical description of state space:

Imagine a space in which you have a lot, but a definite number of discrete points. Each of those points (state vectors) can be described using a combination of base states with appropriate coefficients, provided that the dot/inner product between any two base states (base vectors) i, j is orthogonal (i.e. defined by the Kronecker delta function).

So, what I am wondering is if this is correct by any means. As aforementioned, my exposure to QM is minimal and therefore I may have this completely wrong. Thanks in advance.
 
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  • #2
You're not that far off. The state space is just an ordinary (complex) vector space. It has an infinite number (in fact, a continuum) of points, just like an ordinary vector space, and often has in fact an infinite number of dimensions.

Like any vector space, it has many bases, not all of which are orthonormal, and while its not necessary to take an orthonormal basis, this is almost always done in practice because we usually take as a basis the eigenstates of some collection of observables, and such states are always orthogonal. Of course, since this space doesn't have an obvious geometric interpretation, you should be wondering where the inner product comes from, and a rough answer is that its define so that the last sentence I said is true.
 
  • #3
Thank you for the clarification.
 

What is a rough description of state space?

A rough description of state space is a mathematical concept used in fields such as physics, engineering, and computer science to describe the possible states or values that a system can have. It is often represented as a set of dimensions or variables that define the state of the system.

Why is state space important in scientific research?

State space is important in scientific research because it allows for the mathematical modeling and analysis of complex systems. By defining the possible states of a system, scientists can better understand its behavior and make predictions about how it will respond to different inputs or conditions.

What are some common examples of state space in different fields?

In physics, state space can be used to describe the position and momentum of a particle in motion. In engineering, state space can be used to model the behavior of a mechanical system. In computer science, state space can be used to describe the possible states of a software program.

How is state space related to phase space?

State space and phase space are closely related concepts. While state space describes the possible states of a system, phase space describes the possible trajectories or paths that a system can take through those states. In other words, phase space is a visualization of state space over time.

What are some limitations of using state space in scientific research?

One limitation of state space is that it can quickly become complex and difficult to visualize when working with systems that have many variables or dimensions. Additionally, state space may not always accurately represent the true behavior of a system, as it is a simplified mathematical model. It is important for scientists to be aware of these limitations when using state space in their research.

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