Show that a unitary operator maps one ON-basis to another

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In summary, we are asked to show that the image of an orthonormal basis under a unitary operator is also an orthonormal basis in an inner product space. This can be easily demonstrated by showing that the inner product is preserved under the unitary operator, and since the orthonormality of the original set implies the orthonormality of the transformed set, the transformed set must also form a basis in the inner product space.
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Homework Statement


Given an inner product space [itex]V[/itex], a unitary operator [itex]U[/itex] and a set [itex]\left\{\epsilon_i\right\}_{i=1,2,\dots}[/itex] which is an orthonormal basis of [itex]V[/itex], show that the image of [itex]\left\{\epsilon_i\right\}[/itex] under [itex]U[/itex] is also an orthonormal basis of [itex]V[/itex]

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The Attempt at a Solution


From the definition of a unitary operator we have [itex]U^{\dagger}=U^{-1}[/itex]. From this we can pretty easily demonstrate that the inner product is preserved under [itex]U[/itex], i.e.
[itex]\left\langle a|b\right\rangle=\left\langle a^{\prime}|b^{\prime}\right\rangle\quad\mbox{where }\left|a^{\prime}\right\rangle=U\,\left|a\right\rangle\quad\mbox{and }\left|b^{\prime}\right\rangle=U\,\left|b\right\rangle[/itex]​

"[itex]\left\{\epsilon_i\right\}_{i=1,2,\dots}[/itex] is orthonormal" can be stated as
[itex]\left\langle \epsilon_i|\epsilon_j\right\rangle=\delta_{i,j}[/itex]​
Since [itex]U[/itex] preserves the inner product, if this statement is true for [itex]\left\{\epsilon_i\right\}[/itex] then it must be true for the image under [itex]U[/itex] as well. In other words, if {\epsilon_i\right\} is an orthonormal set, then the image of {\epsilon_i\right\} under [itex]U[/itex] is an orthornomal set as well.

So we have "ortho" and "normal", but where I've hit a mental block is at the "basis" part. For an [itex]N[/itex]-dimensional space, it is clear that any set contained in the space which is orthonormal and has the same number of elements as an ON-basis of the space must also be an ON-basis of the space. What I'm stuck on is how to generalize this to an infinite-dimensional space.
...or am I going about this the wrong way?
 
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If the transformed vectors also form an orthonormal set, then they are automatically linearly independent hence form a basis in the preHilbert space, if they are maximal. Can you show maximality ?
 

1. How do you define a unitary operator?

A unitary operator is a linear transformation that preserves the inner product of vectors. In other words, it preserves the length and angle between vectors. Mathematically, a unitary operator U satisfies U*U = UU* = I, where U* denotes the conjugate transpose of U and I is the identity operator.

2. What is an ON-basis?

An ON-basis (orthonormal basis) is a set of vectors that are orthogonal (perpendicular) to each other and have a norm (length) of 1. In other words, the inner product of any two vectors in an ON-basis is 0 and the norm of each vector is 1.

3. How does a unitary operator map one ON-basis to another?

A unitary operator maps one ON-basis to another by rotating and/or reflecting the vectors in the original basis. This is done while preserving the length and angle between vectors, ensuring that the resulting basis is still orthonormal.

4. Why is it important for a unitary operator to map one ON-basis to another?

Unitary operators are important in quantum mechanics because they represent physical transformations such as rotations and reflections, which are fundamental in understanding the behavior of quantum systems. By mapping one ON-basis to another, unitary operators allow us to analyze and manipulate quantum systems in a consistent and mathematically rigorous way.

5. Can a non-unitary operator also map one ON-basis to another?

No, a non-unitary operator cannot map one ON-basis to another. This is because a non-unitary operator does not preserve the length and angle between vectors, which are crucial properties of an ON-basis. Only a unitary operator can transform a basis while maintaining these properties.

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