TISE in the position representation- basic question

In summary, the time independent Schrodinger equation can be applied to wavefunctions in the position representation by multiplying on the left by <x| and inserting the identity operator, resulting in <x|H|x'> = E<x|ψ>, where <x|ψ> is the wavefunction and <x|H|x'> is the Hamiltonian expressed as an operator acting on functions of x. This is valid because operators can operate on functions of space.
  • #1
Lucy Yeats
117
0
We were told in lectures that the time independent Schrodinger equation can be applied to wavefunctions, i.e. [itex]\frac{hbar^2}{2m}[/itex][itex]\frac{d^2U}{dx^2}[/itex]+V(x)U=EU where U is the wavefunction bra x ket psi. I don't understand why this is valid, as wavefunctions are probability amplitudes, and operators can't operate on mere numbers. Could someone explain how this result is derived? In other words, how do you apply the TISE in the position representation?

Thanks in advance for your help! :smile:
 
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  • #2
Wave functions are probability amplitudes, as such they are functions of space (and time). Operators in the position representation operate on functions of space (and time), so I don't see what's the problem.

You apply it like you would apply a (second) derivative and multiplication to any function from regular calculus.
 
  • #3
Thanks for your help.
To rephrase my question:
H/En>=En/En> (where /x> is ket x)
Multiplying by bra x, <x/H/En>=En<x/En>.
I would like to know why <x/H/En>=H<x/En>.
 
  • #4
Abstractly, in terms of bras and kets, H|ψ> = E|ψ>. If we want to write this in the x representation, we multiply on the left by <x|, and also insert the identity operator, in the form I = ∫|x>d3x<x|

∫<x|H|x'>d3x'<x'|ψ> = E<x|ψ>

Then <x|ψ> is the wavefunction ψ(x), and <x|H|x'> is the Hamiltonian expressed as an operator acting on functions of x. For example, the V term is <x|V|x'> = V(x) δ3(x - x').
 
  • #5
Lucy Yeats said:
Thanks for your help.
To rephrase my question:
H/En>=En/En> (where /x> is ket x)
Multiplying by bra x, <x/H/En>=En<x/En>.
I would like to know why <x/H/En>=H<x/En>.

Well, you can convince yourself this way:
[itex]<x| \dfrac{p^2}{2m} |\Psi>[/itex]
[itex]= <x| \dfrac{p^2}{2m} |k><k|\Psi>[/itex]
[itex]= <x| \dfrac{h^2 k^2}{2m} |k><k|\Psi>[/itex]
[itex]= \dfrac{h^2 k^2}{2m} <x|k><k|\Psi>[/itex]
[itex]= \dfrac{h^2 k^2}{2m} \dfrac{1}{2\pi} e^{i kx} <k|\Psi>[/itex]
[itex]= - \dfrac{h^2}{2m} \dfrac{\partial^2}{\partial x^2} \dfrac{1}{2\pi} e^{i kx} <k|\Psi>[/itex]
[itex]= - \dfrac{h^2}{2m} \dfrac{\partial^2}{\partial x^2} <x|k><k|\Psi>[/itex]
[itex]= - \dfrac{h^2}{2m} \dfrac{\partial^2}{\partial x^2} <x|\Psi>[/itex]
 
  • #6
When operator act on a ket state, it produces an eigenvalue which is merely number and commutes with everything. Also, operators have their own positional form, such as momentum operator which we usually know as the partial derivative with respect to space. Written like that, I think it would be more reasonable to act on a function of space.

In addition, wavefunction came up first, and bra-ket are introduced later to be more general rather than specific to positional. Therefore, physicists definitely have ensured that they are equivalent.
 

Related to TISE in the position representation- basic question

What is TISE in the position representation?

TISE stands for the Time-Independent Schrodinger Equation and it is a fundamental equation in quantum mechanics that describes how the wave function of a quantum system changes over time.

What is the position representation in quantum mechanics?

The position representation, also known as the position space representation, is a way of representing the state of a quantum system using the position of the particles in the system. It is one of the most commonly used representations in quantum mechanics.

How does TISE work in the position representation?

In the position representation, the TISE is written as an equation that relates the position of the particles to their wave function. This equation allows us to solve for the wave function and determine the probability of finding the particles at different positions in the system.

What are the basic concepts involved in TISE in the position representation?

The basic concepts involved in TISE in the position representation include the wave function, potential energy, and the Hamiltonian operator. These concepts are used to describe the behavior and evolution of quantum systems in the position representation.

What are some real-world applications of TISE in the position representation?

TISE in the position representation has many applications in fields such as quantum chemistry, solid-state physics, and materials science. It is used to study the behavior of atoms, molecules, and materials at the quantum level, and has helped in the development of new technologies such as transistors and lasers.

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