A quantum time-dependent spectrum analysis, or simply, a quantum spectral analysis (QSA) is presented in this work based on Schrödinger equation, which is a partial differential equation that describes how the quantum state of a non-relativistic physical system changes with time. In the classic world, it is named frequency in time (FIT), which is presented here in opposition and as a complement of traditional spectral analysis frequency-dependent based on Fourier theory. Besides, FIT is a metric, which assesses the impact of the flanks of a signal on its frequency spectrum, which is not taken into account by Fourier theory, let alone in real time. Even more, and unlike all derived tools from Fourier Theory, (i.e., continuous, discrete, fast, short-time, fractional and quantum Fourier Transform, as well as, Gabor) FIT has the following advantages among many others: a) compact support with excellent energy output treatment, b) low computational cost, O(N) for signals and O(N2) for images, c) does not have phase uncertainties (indeterminate phase for magnitude = 0) as Discrete and Fast Fourier Transform has (DFT, FFT, respectively). In fact, FIT constitutes one side of a triangle (which from now on is closed) which consists of the original signal in time, the spectral analysis based on Fourier Theory and FIT. Thus a toolbox is completed, which is essential for all applications of Digital Signal Processing (DSP) and Digital Image Processing (DIP); and, even, in the latter, FIT allows edge detection (which is called flank detection in case of signals), denoising, despeckling, compression, and superresolution of still images. Such applications will be extremely useful for signals, imagery and communication intelligence.