Stationary States: Differential Equations Issues

In summary, the conversation discusses an equation from Griffith's textbook that includes the constant h-bar and how it can be used to determine the value of V in a simplified case. The conversation also explores the importance of setting the equation equal to a constant and how it allows for simpler mathematical calculations. The expert suggests finding a function that is not a constant but satisfies both sides of the equation for all allowable values of x and t.
  • #1
Elwin.Martin
207
0
So in Griffith's (ed. 2 page 37) there's an equation that says that
**pretend the h's are h-bars...I don't know Latex very well**

ih[itex]\frac{1}{\varphi}[/itex][itex]\frac{d\varphi}{dt}[/itex]=-[itex]\frac{h^{2}}{2m}[/itex][itex]\frac{1}{\psi}[/itex][itex]\frac{d^{2}\psi}{dx^{2}}[/itex]+V

Since in this simplified case V where is a function of x alone he says that each side is equal to a constant but I'm still trying to figure out why.

I can see that the LHS is a function of t alone because he made the wave function separable and the [itex]\varphi[/itex] is a function of t and the [itex]\psi[/itex] is a function of x but I'm not sure why it's important. I see that if we set the equation equal to a constant the rest of the math works out nicely but I can't see what allows us to do this.

If we had say [itex]\frac{dy}{dx}[/itex]=[itex]\frac{dz}{dt}[/itex] and we set the whole thing equal to c, how would I know that y(x) = cx and z(t) = ct (they're the same constant right?) so y(x)/x=z(t)/t ?


Any direction would be great and thank you for your time,
elwin.
 
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  • #2
Elwin.Martin said:
**pretend the h's are h-bars...I don't know Latex very well**

h-bar in Latex is \hbar (surprise! ).

The equation has to be true for any value of x and any value of t.

First pick a value of t and use it to evaluate the LHS. This gives you a number. No matter what value you use for x, the RHS must evaluate to that number. So the RHS is constant.

Now start over. Pick a value of x and use it to evaluate the RHS... (see if you can fill in the rest.)
 
  • #3
Can you see another possibility? Remember that mathematically, a variable is free to take any value in its domain. So if the LHS is only a function of t, and the RHS is only a function of x, can you find any function f(x,t) that is NOT a constant, yet still satisfies both sides of the equation independently?

In other words, what you are given is:

g(x)=h(t)

and I am asking you to do is find any f(x,t) that is NOT a constant and satisfies BOTH

g(x)=f(x,t)

AND

h(t)=f(x,t)

for ALL allowable values of x and t.
 
  • #4
jtbell said:
h-bar in Latex is \hbar (surprise! ).

The equation has to be true for any value of x and any value of t.

First pick a value of t and use it to evaluate the LHS. This gives you a number. No matter what value you use for x, the RHS must evaluate to that number. So the RHS is constant.

Now start over. Pick a value of x and use it to evaluate the RHS... (see if you can fill in the rest.)

haha thank you for that

I think I see what you mean now.
 
  • #5
SpectraCat said:
Can you see another possibility? Remember that mathematically, a variable is free to take any value in its domain. So if the LHS is only a function of t, and the RHS is only a function of x, can you find any function f(x,t) that is NOT a constant, yet still satisfies both sides of the equation independently?

In other words, what you are given is:

g(x)=h(t)

and I am asking you to do is find any f(x,t) that is NOT a constant and satisfies BOTH

g(x)=f(x,t)

AND

h(t)=f(x,t)

for ALL allowable values of x and t.

That makes a whole lot more sense!

thank you very much for your time,
elwin.
 

FAQ: Stationary States: Differential Equations Issues

1. What are stationary states in differential equations?

Stationary states refer to solutions of a differential equation that do not change over time. In other words, they are equilibrium points where the derivative of the function is equal to zero.

2. How are stationary states determined in differential equations?

To determine stationary states, the differential equation is set equal to zero and solved for the variables. This results in a set of values for the variables that represent the equilibrium points.

3. What is the significance of stationary states in differential equations?

Stationary states provide important information about the behavior of a system described by a differential equation. They can help determine stability and critical points, and can be used to analyze the long-term behavior of the system.

4. Can a system have multiple stationary states?

Yes, a system can have multiple stationary states. In fact, most systems have more than one stationary state, and the number and location of these states can greatly affect the behavior of the system.

5. How are stationary states related to stability in differential equations?

In general, stationary states that are stable indicate that the system will remain near that state over time. Unstable stationary states, on the other hand, suggest that the system will not stay at that state and will instead move away from it. This is an important consideration in analyzing the behavior of a system.

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