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*As mentioned, quantum mechanics is based on operators. In classical physics, quantities such as momentum and energy are usually thought of as scalars. Say you have a free classical particle of mass m travelling at a velocity v starting from some initial position a. What is its energy? Well, if you know that E = K + V, then you can simply write . But what if your particle is quantum? Well, now velocity (or, more commonly stated, momentum) is not well-defined so we can’t give a complete description of the state of our system simply by stating its velocity and momentum. We can, however, write a wave equation that describes the state of our system and operate on it to determine its properties. To start with, consider an electromagnetic wave in one dimension (substitutions were made using the de Broglie relations
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