Mind Network  Samuel Solomon
Quantum Mechanics' Core Postulates
I know that quantum mechanics can be a little frustrating in the beginning because it is based on unprovable postulates. Hopefully, I have given you at least the motivation and thought process that goes into each postulate. It may be useful to know that scientists can propose and reject postulates as the field grows, but as long as we are clearly defining what we assume, no matter how trivial, we can easily go back when a problem arises.
Quantum mechanics was not readily accepted into the physics community, but it remains a powerful predictive tool of our microscopic world. Here are the core postulates we make in quantum mechanics that cannot be mathematically or experimentally shown to be true:
1. Psi(x), the wave function, fully describes the quantum state (every observable property of the particle)
Thought process: There is something about the particle that acts like a wave
2. Psi(x,t)^2 is the probability density with C_n^2 describing the probability of being in the basis state phi_n
Thought process: Once we normalize the function, a higher coefficient weights that wave basis more
3. Physical observables are found from linear, hermitian operators
Thought process: If they were not linear functions then it would not allow for a linear superposition of states
4. Measuring a quantum observable corresponding to one of the basis states collapses psi(x) to that basis function
Experimentally: After measurement, each observable measured will be for that basis 100% of the time
5. The Schrödinger equation can be used to time evolve any time independent quantum state
Thought process: The postulated equation allowed us to focus on one state and add the time term in later on
Remember, we are still building up our quantum mechanical toolbox after tragically throwing out tons of physics from classical mechanics. What can we keep, what can we throw away? Physicists have worked very hard to vigorously (even tediously) prove every detail in our physical world. If some of these postulates make you feel questionable, remember them! Remember what you assume for when you find a problem you cannot solve (it may just lead you somewhere interesting). For sure, this nonrelativistic quantum mechanics cannot explain the universe, but postulates are getting stricter and bumps are being smoothed over. The next iteration of quantum mechanics is quantum field theory (not discussed in this course).
Quantum mechanics was not readily accepted into the physics community, but it remains a powerful predictive tool of our microscopic world. Here are the core postulates we make in quantum mechanics that cannot be mathematically or experimentally shown to be true:
1. Psi(x), the wave function, fully describes the quantum state (every observable property of the particle)
Thought process: There is something about the particle that acts like a wave
2. Psi(x,t)^2 is the probability density with C_n^2 describing the probability of being in the basis state phi_n
Thought process: Once we normalize the function, a higher coefficient weights that wave basis more
3. Physical observables are found from linear, hermitian operators
Thought process: If they were not linear functions then it would not allow for a linear superposition of states
4. Measuring a quantum observable corresponding to one of the basis states collapses psi(x) to that basis function
Experimentally: After measurement, each observable measured will be for that basis 100% of the time
5. The Schrödinger equation can be used to time evolve any time independent quantum state
Thought process: The postulated equation allowed us to focus on one state and add the time term in later on
Remember, we are still building up our quantum mechanical toolbox after tragically throwing out tons of physics from classical mechanics. What can we keep, what can we throw away? Physicists have worked very hard to vigorously (even tediously) prove every detail in our physical world. If some of these postulates make you feel questionable, remember them! Remember what you assume for when you find a problem you cannot solve (it may just lead you somewhere interesting). For sure, this nonrelativistic quantum mechanics cannot explain the universe, but postulates are getting stricter and bumps are being smoothed over. The next iteration of quantum mechanics is quantum field theory (not discussed in this course).

