Can subspaces be used to determine probabilities in quantum mechanics?

In summary, a subspace in quantum mechanics is a subset of a larger vector space that is closed under the operations of the vector space. It is related to quantum states as it represents different aspects of a state, such as spin orientation. Subspaces can be used to simplify calculations, but they have limitations and cannot represent entangled states. Subspaces are also related to observables, as they can be defined by Hermitian operators and their eigenvectors correspond to possible measurement outcomes.
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Suppose we have an observable with a certain number of eigenstates. We would normalize all these possibilities to 1 in order to give each eigenstate an appropriate probability of being measured. Can we then only consider the data of many measurements for only a subset of those eigenstates and normalize that subset to 1 and get different probabilities for considering only that subset of alternatives? Is that subset called a subspace of the original Hilbert space? And can this be done for any arbitrary subset of the original eigenstates?
 
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Yes, yes and yes.
 

1. What is a subspace in quantum mechanics?

A subspace in quantum mechanics refers to a subset of a larger vector space that is closed under the operations of the vector space. In other words, any linear combination of vectors in the subspace will still result in a vector within the subspace.

2. How are subspaces related to quantum states?

In quantum mechanics, a state is represented by a vector in a vector space. This vector space can be broken down into subspaces, each representing a different aspect of the state. For example, the subspace of spin states contains all possible spin orientations of a particle.

3. Can subspaces be used to simplify quantum calculations?

Yes, subspaces can be used to simplify quantum calculations by reducing the number of dimensions needed to represent a system. By breaking a larger vector space into smaller subspaces, the calculations become more manageable and the results can be easily interpreted.

4. Are there any limitations to using subspaces in quantum mechanics?

While subspaces can be useful for simplifying calculations, they do have limitations. For example, subspaces cannot be used to represent entangled states, where the state of one particle depends on the state of another particle. In these cases, the entire vector space must be used.

5. How are subspaces related to observables in quantum mechanics?

In quantum mechanics, observables are represented by Hermitian operators. These operators can be used to define subspaces within a vector space. The eigenvectors of the operator, which are vectors within the subspace, correspond to the possible measurement outcomes of the observable.

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