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We consider rotationally invariant noncommutative algebra with tensors ofnoncommutativity constructed with the help of additional coordinates andmomenta. The algebra is equivalent to well known noncommutative algebra ofcanonical type. In the noncommutative phase space with rotational symmetryinfluence of noncommutativity on the spectrum of free particle and spectrum ofharmonic oscillator is studied up to the second order in the parameters ofnoncommutativity. We find that because of momentum noncommutativity thespectrum of free particle is discrete and corresponds to the spectrum ofharmonic oscillator in the ordinary space (space with commutative coordinatesand commutative momenta). We obtain the spectrum of the harmonic oscillator inthe rotationally invariant noncommutative phase space and conclude thatnoncommutativity of coordinates affects on its mass. The frequency of theoscillator is affected by the coordinate noncommutativity and the momentumnoncommutativity. On the basis of the results, the eigenvalues of squaredlength operator are found and restrictions on the value of length innoncommutative phase space with rotational symmetry are analyzed.

Effect of noncommutativity on the spectrum of free particle and harmonic oscillator in rotationally invariant noncommutative phase space