Calculating Probabilities in Quantum Mechanics revisited

[Alan Grayson]

Since my graduate notes are presently inaccessible, I will likely have some errors in what I am about to write. IIUC, we can use S's eqn with known boundary conditions to calculate the wf for a particular system. This wf is a mathematical function. Then, to get a probability for measuring a particular eigenvalue, we must take the inner product of this function with another function, the superposition of states, to get a real value less than one. But since this superposition has unknown, complex, multiplicative factors for each eigenfunction in its sum, how can we get a probability value from this procedure? How can the possible eigenvalues be determined? What is my misconception in the process of calculating probabilities in QM? TY, AG

[Alan Grayson]

On Thursday, July 31, 2025 at 11:21:28 PM UTC-6 Alan Grayson wrote:

Since my graduate notes are presently inaccessible, I will likely have some errors in what I am about to write. IIUC, we can use S's eqn with known boundary conditions to calculate the wf for a particular system. This wf is a mathematical function. Then, to get a probability for measuring a particular eigenvalue, we must take the inner product of this function with another function, the superposition of states, to get a real value less than one. But since this superposition has unknown, complex, multiplicative factors for each eigenfunction in its sum, how can we get a probability value from this procedure? How can the possible eigenvalues be determined? What is my misconception in the process of calculating probabilities in QM? TY, AG

I will check Dirac's book on QM this evening to determine how it's done. AG 

[John Clark]

On Fri, Aug 1, 2025 at 11:15 PM Alan Grayson <agrays...@gmail.com> wrote:

> we can use S's eqn with known boundary conditions to calculate the wf for a particular system. This wf is a mathematical function.

OK

> Then, to get a probability for measuring a particular eigenvalue, we must take the inner product of this function with another function, the superposition of states, to get a real value less than one. But since this superposition has unknown, complex, multiplicative factors for each eigenfunction in its sum, how can we get a probability value from this procedure? 

 

If as you say we can "calculate the wf for a particular system" then we must know the complex, multiplicative factors that make up that wave function, and we would know them if an electron had passed through a Stern–Gerlach magnet, or a photon passed through a polarizing filter. 

If you don't know those complex, multiplicative factors then Schrodinger's equation cannot help you, but Quantum Mechanics is hardly unique in that regard. Newton can't tell you where a baseball is going to be in 3 seconds unless you tell him where the baseball is now and what its velocity is.

John K Clark    See what's on my new list at  Extropolis

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