Can Scalars Represent Quantum States Effectively?

[tenchotomic]

Title may sound weird,but I think it might be worth exploring

In axiomatic formulation of quantum mechanics, quantum states are postulated as vectors residing in Hilbert space.
The only apriori requirement that Iam aware of ,for a quantity to qualify as a quantum state, is that it should obey principle of superposition ,which a scalar quantity would obey as good as a vector.
Then why don't we take quantum states as scalars?

What I think could be the reason:
Quantum states are characterized by certain dynamical variables
In standard formalism we represent dynamical variables as operators acting as linear transformation on vectors.
Now,scalars are not operated on by operators,so by taking scalars as quantum states we can no longer characterize a quantum state by physically observable properties,which would render it meaningless.

I would really like to know whether my answer is correct.

 

[strangerep]

I would really like to know whether my answer is correct.

No, not even close -- sorry.

In axiomatic formulation of quantum mechanics, quantum states are postulated as vectors residing in Hilbert space.
The only apriori requirement that I am aware of ,for a quantity to qualify as a quantum state, is that it should obey principle of superposition ,which a scalar quantity would obey as good as a vector.
Then why don't we take quantum states as scalars?

States as vectors in the Hilbert space correspond to pure states only.
For mixed states, one must use a density matrix (aka state operator).
State operators must satisfy 3 requirements (trace=1, self-adjoint, and non-negative).

What I think could be the reason:
Quantum states are characterized by certain dynamical variables
In standard formalism we represent dynamical variables as operators acting as linear transformation on vectors.
Now,scalars are not operated on by operators,so by taking scalars as quantum states we can no longer characterize a quantum state by physically observable properties,which would render it meaningless.

I would really like to know whether my answer is correct.

Dynamical variables correspond to certain operators on the Hilbert space.

(IMHO, you really need a QM textbook. If you can get a copy of Ballentine,
this is explained pretty well in chapters 1-3.)

 

[tenchotomic]

No, not even close -- sorry.



States as vectors in the Hilbert space correspond to pure states only.
For mixed states, one must use a density matrix (aka state operator).
State operators must satisfy 3 requirements (trace=1, self-adjoint, and non-negative).




Dynamical variables correspond to certain operators on the Hilbert space.

(IMHO, you really need a QM textbook. If you can get a copy of Ballentine,
this is explained pretty well in chapters 1-3.)

Yeah,Thanks for advice.But I guess I haven't got my answer yet (iff question makes some sense ofcourse)

 

[strangerep]

But I guess I haven't got my answer yet (iff question makes some sense ofcourse)

I'll try a different tack...

Scalars are 1-dimensional vectors. In QM, multiplying a state by a scalar does not change the physical situation. I.e., physical states really correspond to rays in Hilbert space, not merely vectors. So if you use scalars only, then you only have one physical state -- which would not be very useful.

 

[tenchotomic]

I'll try a different tack...

Scalars are 1-dimensional vectors. In QM, multiplying a state by a scalar does not change the physical situation. I.e., physical states really correspond to rays in Hilbert space, not merely vectors. So if you use scalars only, then you only have one physical state -- which would not be very useful.

Thanks for your view.