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E.g., for atomic transitions you have in the electric-dipole approximation

$$\hat{H}'(t) \propto \vec{E}(t) \cdot \sum_k q_k \hat{\vec{x}}_k,$$

which is not a scalar when considering only the degrees of freedom of the atom. This is intuitively clear, because ##\vec{E}(t)## defines a preferred direction.

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Thank you for this! I am still a bit confused. I thought that the Hamiltonian must always be a scalar (I've seen this statement in many QM readings). So it is not always a scalar?

E.g., for atomic transitions you have in the electric-dipole approximation

$$\hat{H}'(t) \propto \vec{E}(t) \cdot \sum_k q_k \hat{\vec{x}}_k,$$

which is not a scalar when considering only the degrees of freedom of the atom. This is intuitively clear, because ##\vec{E}(t)## defines a preferred direction.

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Look here:Thank you for this! I am still a bit confused. I thought that the Hamiltonian must always be a scalar (I've seen this statement in many QM readings). So it is not always a scalar?

https://www.physicsforums.com/threads/lagrangian-vs-hamiltonian-in-qft.658142/

I believe you are confusing between the physicist's meaning of the word "scalar" and the mathematicians' use of this word.

For physicists a scalar is: "Formally, a scalar is unchanged by coordinate system transformations", where in maths a scalar is just an element in the field which the vector space is defined over.

In the example vanhees gave you, I assume the Hamiltonian is a pseudo-scalar, and in after a suitable change of coordinates we get a minus sign there.

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I believe you are confusing between the physicist's meaning of the word "scalar" and the mathematicians' use of this word.

Really? I've seen both physicists and mathematicians using word 'scalar' in both of this meanings. It just depends on the context.

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Of course, if you transform both ##\vec{E}## and ##\vec{x}## it's a scalar, but here you consider ##\vec{E}## as an external (classical) field, and thus you break full rotation symmetry to symmetry under rotations of ##\vec{x}## around ##\vec{E}##.Thank you for this! I am still a bit confused. I thought that the Hamiltonian must always be a scalar (I've seen this statement in many QM readings). So it is not always a scalar?

This is not different from classical mechanics!

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