Undergrad Momentum operator in quantum mechanics

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SUMMARY

The momentum operator in quantum mechanics is defined as -iħd/dx for one spatial dimension and -iħ∇ for three spatial dimensions, where the latter is a vector operator. In three dimensions, momentum is represented as a polar vector, necessitating the use of a vector operator in quantum mechanics. This operator transforms correctly as a vector when applied to scalar fields, such as the Schrödinger wave function, which is a scalar field under rotations. The discussion clarifies that while 3D operators can appear as scalar operators in 1D, this is not unique to quantum mechanics.

PREREQUISITES
  • Understanding of quantum mechanics fundamentals
  • Familiarity with vector calculus
  • Knowledge of the Schrödinger equation
  • Basic concepts of classical mechanics and canonical momentum
NEXT STEPS
  • Study the mathematical formulation of the momentum operator in quantum mechanics
  • Explore the implications of vector operators in quantum mechanics
  • Learn about the transformation properties of quantum operators under rotations
  • Investigate the relationship between classical and quantum momentum representations
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Quantum mechanics students, physicists, and researchers interested in the mathematical foundations of momentum operators and their applications in quantum theory.

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The momentum operator for one spation dimension is -iħd/dx (which isn't a vector operator) but for 3 spatial dimensions is -iħ∇ which is a vector operator. So is it a vector or a scalar operator ?
 
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In 3D the momentum is a (polar) vector and thus must be represented by a vector operator in quantum mechanics, and that's indeed the case as you correctly wrote: In the position representation,
$$\hat{\vec{p}}=-\mathrm{i} \hbar \vec{\nabla},$$
and this transforms indeed as a vector when operating on scalar fields (and the Schrödinger wave function is a scalar field under rotations!).
 
Your objection, that 3-d operators in 3-d can appear as scalar operators in 1-d, isn't really specific to QM.
 
Well, technically, as a covector :P
 
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That's no quantum specific. Also in classical physics (canonical) momenta are co-vectors.
 
I know. I'm in a pedantic mood ;)
 
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Thread 'Two equivalent statements of time reversal symmetric Hamiltonian'

Time reversal invariant Hamiltonians must satisfy ##[H,\Theta]=0## where ##\Theta## is time reversal operator. However, in some texts (for example see Many-body Quantum Theory in Condensed Matter Physics an introduction, HENRIK BRUUS and KARSTEN FLENSBERG, Corrected version: 14 January 2016, section 7.1.4) the time reversal invariant condition is introduced as ##H=H^*##. How these two conditions are identical?
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