Inspired by a sample question in the textbook of quantum mechanics，the spatial correlations of two identical particles in the one-dimensional harmonic oscillator potential are discussed，where the exchange symmetry of wave function is assumed to be classical，symmetrical and anti-symmetrical，respectively. It is found that the results for free particles still hold for the one -dimensional harmonic oscillator bound states，namely，when the exchange symmetry is symmetrical，the two particles tend to draw close each other，the probability of two particles approaching each other( r→0) tends to be zero while when anti-symmetrical，which will be useful for understanding the quantum effects induced by the exchange symmetry.