How Does the Quantum Operator \(\hat{p}^2\) Derive from \(\hat{p}\)?

In summary, the given equation \hat{p} = -i\hbar (\frac{\partial}{\partial r} + \frac{1}{r}) can be used to show that \hat{p}^2 = -\frac{\hbar^2}{r^2} \frac{\partial}{\partial r}(r^2 \frac{\partial}{\partial r}) by operating on a test function with the operator \hat{p} and then taking it away.
  • #1
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Homework Statement


Given that [itex] \hat{p} = -i\hbar (\frac{\partial}{\partial r} + \frac{1}{r}) [/itex], show that [itex] \hat{p}^2 = -\frac{\hbar^2}{r^2} \frac{\partial}{\partial r}(r^2 \frac{\partial}{\partial r}) [/itex]

Homework Equations



Above

The Attempt at a Solution


I tried [itex]\hat{p}\hat{p} = -\hbar^2((\frac{\partial}{\partial r})^2 + \frac{1}{r} \frac{\partial}{\partial r} + \frac{\partial}{\partial r}\frac{1}{r} +\frac{1}{r^2}) [/itex].

This gave me [itex] -\hbar^2((\frac{\partial}{\partial r})^2 + \frac{1}{r} \frac{\partial}{\partial r} )[/itex] instead of the 2 / r factor I needed.
 
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  • #2
Nope. ##{\partial \over \partial r}{1\over r} ## gives ##{1\over r}{\partial \over \partial r} -{1\over r^2}##
Remember p is an operator: you have to imagine there is something to the right of it to operate on.
 
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  • #3
BvU said:
Nope. ##{\partial \over \partial r}{1\over r} ## gives ##{1\over r}{\partial \over \partial r} -{1\over r^2}##
Remember p is an operator: you have to imagine there is something to the right of it to operate on.

Thanks!

I suppose it makes it easier if I had used a test function, and then taken it away.
 

Related to How Does the Quantum Operator \(\hat{p}^2\) Derive from \(\hat{p}\)?

What is a quantum mechanics operator?

A quantum mechanics operator is a mathematical representation of a physical observable in quantum mechanics, such as position, momentum, or energy. It acts on the wavefunction of a quantum system to yield a measurable value.

How do quantum mechanics operators work?

Quantum mechanics operators work by operating on the wavefunction of a quantum system, transforming it into a new wavefunction that represents the system after the observable has been measured. The result of the operation is a probability distribution of possible outcomes for that observable.

What are the types of quantum mechanics operators?

There are three main types of quantum mechanics operators: Hermitian operators, unitary operators, and anti-Hermitian operators. Hermitian operators represent physical observables and have real eigenvalues, while unitary operators represent transformations and preserve the norm of the wavefunction. Anti-Hermitian operators have imaginary eigenvalues and represent anti-unitary transformations.

How are quantum mechanics operators represented mathematically?

Quantum mechanics operators are represented by matrices in a mathematical formalism known as matrix mechanics. The elements of the matrix correspond to the possible outcomes of the observable being measured. Alternatively, in wave mechanics, operators are represented by differential equations known as operators.

What is the significance of quantum mechanics operators in quantum computing?

Quantum mechanics operators play a crucial role in quantum computing, as they are used to manipulate and measure the quantum states of qubits. Operators such as the Pauli matrices and the Hadamard gate are used in quantum algorithms to perform operations on qubits, leading to the potential for exponentially faster computation.

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