Proper form of schrodinger's equation?

In summary, the first form is incorrect because it uses a partial derivative instead of a total derivative.
  • #1
unchained1978
93
0
I feel a bit silly asking this, but I've been working through some QM lately and there's one aspect of Schrodinger's equation that's puzzling me. I've typically understood the equation as [itex]i\hbar \frac{d|\psi\rangle}{dt}=\hat H |\psi\rangle[/itex], but I've also seen it written as [itex]i\hbar \frac{\partial |\psi\rangle}{\partial t}=\hat H|\psi\rangle[/itex]. The use of a partial derivative instead of a total derivative is what's got me. In most cases, I know it doesn't matter but I can imagine some where it would. The total derivative indicates that [itex]\psi[/itex] may not explicitly depend on time, but implicitly could through some other variable which is dependent on time. This evolution is generated by the Hamiltonian. In the other case, the partial derivative only concerns explicit time dependence, and hence this form only makes sense to me when there is an explicit [itex]t[/itex] showing up in the equations. Which form is generally correct?
 
Physics news on Phys.org
  • #2
The form is telling you something about the nature of the state vector isn't it?
The first is either sloppy notation of the state vector depends on time alone (i.e. not space).
In general, a wavefunction or vector will depend on time and space and so you should use the partial notation.
http://www.nyu.edu/classes/tuckerman/stat.mech/lectures/lecture_12/node5.html
 
  • #3
Since you've written the SE in the Dirac bra/ket notation, and in the standard formulation time is a mere parameter, you can consider using the total derivative symbol d as correct, sind the SE is about time evolution of kets and nothing more.
 
  • #4
Yes, one should really clearly say that in the bra-ket notation [itex]|\psi(t) \rangle[/itex] does not depend on position, momentum, or any other values of observables. It's (in the here used Schrödinger picture of the time evolution) a function of time only, and thus you should use the total differential.

The wave function is always the (generalized) scalar product of a (generalized) common eigenvector of a complete set of compatible observables (e.g., position). In the Schrödinger picture observables are represented by time-independent self-adjoint operators, and thus also the (generalized) eigenvectors are time-independent. Thus you have
[tex]\psi(t,\vec{x})=\langle \vec{x}|\psi(t).[/tex]
Then you have to use the partial derivative to write
[tex]\mathrm{i} \hbar \partial_t \psi(t,\vec{x}) = \langle \vec{x}|\mathrm{i} \hbar \mathrm{d}_t \psi(t) \rangle = \langle \vec{x} |\mathbf{H} \psi \rangle = \hat{H} \langle \vec{x}|\psi \rangle=\hat{H} \psi(t,\vec{x}).[/tex]
Here, [itex]\mathbf{H}[/itex] is the Hamilton operator in abstract Hilbert space and [itex]\hat{H}[/itex] in the position representation.

It is quite important to distinguish the abtract Hilbert-space objects from their representation wrt. to a given (generalized) basis!
 

1. What is the proper form of Schrodinger's equation?

The proper form of Schrodinger's equation is a fundamental equation in quantum mechanics that describes the time evolution of a quantum system. It is written as iħ∂ψ/∂t = Ĥψ, where Ĥ is the Hamiltonian operator, ħ is the reduced Planck's constant, ψ is the wave function, and t is time.

2. Why is the Schrodinger's equation important?

Schrodinger's equation is important because it is the cornerstone of quantum mechanics and allows us to mathematically describe the behavior of microscopic particles. It has been successfully used to predict the behavior of atoms, molecules, and other quantum systems, and has led to many technological advancements, such as transistors and lasers.

3. What are the key principles behind Schrodinger's equation?

Schrodinger's equation is based on two key principles: the wave-particle duality of matter and the superposition principle. The wave-particle duality states that particles can exhibit both wave-like and particle-like behavior, and the superposition principle states that a quantum system can exist in multiple states at the same time.

4. How is Schrodinger's equation different from classical mechanics?

Schrodinger's equation differs from classical mechanics in that it describes the behavior of microscopic particles, while classical mechanics is used to describe the behavior of macroscopic objects. Additionally, classical mechanics is based on deterministic principles, while Schrodinger's equation incorporates probabilistic elements due to the wave-like nature of particles.

5. Can Schrodinger's equation be solved analytically?

In most cases, Schrodinger's equation cannot be solved analytically, meaning that there is no closed-form solution. Instead, numerical methods and approximations are used to solve the equation and obtain useful information about the system. However, there are some simple systems, such as the harmonic oscillator, for which analytical solutions do exist.

Similar threads

Replies
2
Views
228
Replies
17
Views
1K
Replies
7
Views
464
Replies
3
Views
777
Replies
2
Views
541
Replies
5
Views
1K
  • Quantum Physics
Replies
9
Views
790
Replies
5
Views
774
  • Quantum Physics
Replies
4
Views
726
  • Quantum Physics
Replies
19
Views
1K
Back
Top