This is a simple idea, even if it seems abstract at first glance. We must distinguish the liquidator (a mathematical operation) from the object (a function) it acts on. For example, we take the derivative represented by d/dx as the "operator" of mathematics, and suppose it acts on a function, such as x2. The result of this operation is a new function, in this case "2xx". However, some functions have a special property when calculating derivatives. For example, the derivative of "e3x" is "3e3x": here we go back to the original function and multiply it by only one number-here is 3. A function that can only be restored after a given operator acts is called the "eigenfunction" of this operator, and the number multiplied by the eigenfunction after the operator acts is the "eigenvalue" of this operator.
Therefore, for each operator, there is a set and a numerical "library" corresponding to it. This collection forms its "spectrum" When the eigenvalue forms a discrete sequence, the spectrum is "discrete". For example, if there is an operator whose eigenvalues are all integers 0, 1, 2, … the frequency spectrum can also be continuous-for example, when it consists of all numbers between 0 and 1.
The basic concept of quantum mechanics can be expressed as follows: all physical quantities in classical mechanics have an operator in quantum mechanics, and the value that this physical quantity can take is the eigenvalue of this operator. It is important that there is a difference between the concept of physical quantity (expressed by operators) and its concept of quantity (expressed by eigenvalues of operators). In particular, energy is now represented by Hamiltonian, and the observed value of energy level-energy will be represented by the eigenvalue corresponding to this operator.