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Schrödinger Wave Equation

SCHROEDINGER WAVE EQUATION

ˆ H ψ ( x , t ) = i 

∂ψ ( x , t )

∂ t

(both images from

Wikipedia.com)

Schrödinger Wave Equation

ˆ H ψ ( x , t ) = i 

∂ψ ( x , t )

∂ t

Schrödinger Wave Equation:

More-or-less “god-given”, just like Newton’s law F(x) = d p /d t.

It works, and if we can show that it fails, we’ll refine or discard.

You learned in PH425 that the solution to the time-dependent SE is,

in terms of the eigenstates of the Hamiltonian:

ψ (^) ( x , t ) = ϕ

n

n

∑ ( x ) e

− iE n

t / 

ψ (^) ( t ) = ϕ n

e

− iE n

t / 

n

∑

H ϕ

n

= E

n

n

H ϕ

n

( x ) =^ E n

n

( x )

where

where

Notice the parallels to the rope problem we solved last week?

Schrödinger Wave Equation

ψ ( x , t ) = c n

( t )φ n

( x )

n

∑

What follows is cast in terms of wave functions to give you

practice, but in fact the ket notation can be used just as easily.

The ket version of this discussion is in your text (Ch. 3) and

was discussed in PH425.

Expand wave function in energy basis; time dependence

comes from coefficients:

Substitute

ˆ H ψ ( x , t ) = i 

∂ψ ( x , t )

∂ t

ˆ H c n

( t )φ n

( x )

n

∑

= i 

∂

∂ t

c n

( t )^ φ n

( x )

n

∑

c n

( t ) E n

φ n

( x )

n

∑

= i 

∂

∂ t

c n

( t )^ φ n

( x )

n

∑

Schrödinger Wave Equation

ψ ( x , t ) = c n

( t )^ φ n

( x )

n

∑

Expand wave function in energy basis; time dependence

comes from coefficients:

Substitute

c n

( t ) E n

φ n

( x )

n

∑

= i 

∂

∂ t

c n

( t )φ n

( x )

n

∑

−

iE n



⎛

⎝

⎜

⎞

⎠

⎟

c n

( t )φ n

( x ) =

∂ c n

( t )

∂ t

φ n

( x )

c n

( t ) =^ e

−

iE n



t

ψ ( x , t ) = e

−

iE n



t

φ n

( x )

n

∑

Schrödinger Wave Equation

There are usually many functions that

solve the eigenvalue problem, each

function with its own associated

eigenvalue. Label them with index n.

H ϕ

n

( x ) = E n

n

( x )

ˆ H ϕ n

= E n

ϕ n

Ψ( x , t ) = ϕ n

( x ) Ae

− i

E n



t

n

∑

In general, the state of particle is NOT an eigenstate of the

Hamiltonian, but a superposition of eigenstates …

Very important that the energy in the exponent

matches the corresponding eigenstate.

Schrödinger Wave Equation

Example: V ( x ) = 0 (free particle)

H ϕ

n

( x ) = E n

n

( x )

ϕ( x ) = C ' e

i

2 mE



x

• Ce

− i

2 mE



x

1st term is traveling wave (to right) and 2nd is traveling to

left. Same eigenvalue E - “degenerate”

−



2

2 m

d

2

ϕ( x )

dx

2

= E ϕ( x )

ψ ( x , t ) = C ' e

i

2 mE



x −

E



t

⎛

⎝

⎜

⎞

⎠

⎟

• Ce

− i

2 mE



x +

E



t

⎛

⎝

⎜

⎞

⎠

⎟

Schrödinger Wave Equation

Now solve the TISE for different potentials and find

examples where there are many solutions with different

energies

-a a → x

↑

Energy

0

V 0

E

V (^) ( x ) =

V 0

x > a

0 x < a

⎧

⎨

⎪

⎩

⎪

Schrödinger Wave Equation

• Time dependent Schrödinger equation
• Energy eigenvalue equation (time independent SE)
• Eigenstates
• Time dependence
• (Connection to separation of variables)
• Mathematical representations of the above

SCHROEDINGER WAVE EQUATION

REVIEW

Schrödinger Wave Equation

H ψ( x , t ) = i 

∂ψ( x , t )

∂ t

What shall we call the separation constant?

H ϕ( x ) T ( t ) = i 

∂ϕ( x ) T ( t )

∂ t

T ( t )

H ϕ( x ) = i ϕ( x )

∂ T ( t )

∂ t

ϕ( x )

H ϕ( x ) = i 

T ( t )

∂ T ( t )

∂ t

ϕ( x )

H ϕ( x ) = i 

T ( t )

∂ T ( t )

∂ t

= E

Schrödinger Wave Equation

Eigenvalue problem! Solutions (and there

are many in general) depend on particular

V ( x ). E represents the energy of a particle

in that particular eigenstate, and there are

many E values.

H ϕ( x ) = E ϕ( x )

∂ T ( t )

∂ t

= − i

E

T ( t )

H ϕ = E ϕ

T ( t ) = Ae

− i

E



t

T ( t ) = Ae

− i ω t

E

Most general solution is:

ψ ( x , t ) = ϕ

n

n

( x ) e

− iE n

t / 