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- What does the following sentence mean: the matrix representation of operators are basis dependent but the operator itself is bases independent.

I have a question about operators in finite dimension Hilbert space.

I will describe the context before asking the question.

Assume we have two quantum states [itex]| \Psi_{1} \rangle [/itex] and [itex] | \Psi_{2} \rangle [/itex].

Both of the quantum states are elements of the Hilbert space, thus [itex]| \Psi_{1} \rangle [/itex], [itex] | \Psi_{2} \rangle [/itex] [itex]\in[/itex] [itex]H[/itex].

Both quantum states are described in different basis. We can write:

[tex]|\Psi_{1}\rangle = \sum_{i = 1}^{n}a_{i}| \psi_{i}\rangle [/tex] and [tex]|\Psi_{2}\rangle = \sum_{i = 1}^{n}b_{i}| \psi_{i}\rangle [/tex].

The coëfficients, [itex]a_{i}[/itex] or [itex]b_{i}[/itex] are basis dependent.

Next, the dictation tells us that the matrix representation of operators depend on bases but that the operator itself is bases independent.

My question what does this mean, the matrix representation of operators are bases dependent but the operator itself is bases independent.

I will describe the context before asking the question.

Assume we have two quantum states [itex]| \Psi_{1} \rangle [/itex] and [itex] | \Psi_{2} \rangle [/itex].

Both of the quantum states are elements of the Hilbert space, thus [itex]| \Psi_{1} \rangle [/itex], [itex] | \Psi_{2} \rangle [/itex] [itex]\in[/itex] [itex]H[/itex].

Both quantum states are described in different basis. We can write:

[tex]|\Psi_{1}\rangle = \sum_{i = 1}^{n}a_{i}| \psi_{i}\rangle [/tex] and [tex]|\Psi_{2}\rangle = \sum_{i = 1}^{n}b_{i}| \psi_{i}\rangle [/tex].

The coëfficients, [itex]a_{i}[/itex] or [itex]b_{i}[/itex] are basis dependent.

Next, the dictation tells us that the matrix representation of operators depend on bases but that the operator itself is bases independent.

My question what does this mean, the matrix representation of operators are bases dependent but the operator itself is bases independent.