In classical physics, we are used to calculating an object's momentum by multiplying its mass by its velocity. But how do we deal with momentum in Quantum Mechanics, where we commonly deal with wave functions?
A wave function is a mathematical function that contains all the information we can know about any system we're studying. For example, if our system is a single electron, the wave function can be used to calculate the probability of finding the electron at different positions in space, or with different values of momentum.
It's only when we make a measurement on the particle, that we cause a collapse in the wave function, and thus know a certain value for the particle's momentum. So a question we can ask is, how do we mathematically deal with the idea of making a measurement in quantum mechanics?
Since the system is described by the wave function, we apply a "measurement operator" to the wave function, which is the real-life equivalent of making a measurement. Mathematically, this is like applying a matrix to a vector (where the matrix is the operator, and the vector is the wave function). On the right hand side of the equation, we get the "eigenvalue", which is the actual measured value that we find as a result of our experiment. This equation is described as the eigenvalue equation.
But what does the momentum operator, that we use in the eigenvalue equation, actually look like? Does it look similar to mass x velocity, which is what momentum looks like in classical physics? The answer is no - the momentum operator is more complex, with the imaginary number, reduced Planck constant, and a partial derivative with respect to x as part of the expression.
The reason for this, is that the operator is derived by considering a "building block" wave function (which is sinusoidal). This can be written in terms of the exponential function, and the position and momentum of the particle. This building block function can also be used to break down almost any other function we deal with. When we differentiate this building block wave function, we find that the result is equal to the momentum, p, of the particle, multiplied by the original wave function and some other factors. Thus if we rearrange this, we find the momentum operator expression.
It's worth noting though that the logic behind momentum is still the same - the conservation of momentum still applies, and the idea is still that it is the mass of the object multiplied by its velocity.
More information about the exponential function with the imaginary number in the exponent: en.wikipedia.org/wiki/Euler%2...
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Timestamps:
0:00 - Classical Momentum (p = mv) and Wave Functions
2:36 - Mathematically Encoding Momentum in Quantum Mechanics
3:44 - What Does the Momentum Measurement Operator Look Like, and Where Does it Come From?
5:33 - Breaking Down Complex Wave Functions into Simple Building Blocks
6:17 - Deriving the Momentum Operator by Differentiating the Wave Function Building Block
1 авг 2024
