Introduction to Quantum Mechanics II

Intro to Quantum Mechanics - part II

The wavefunction

Background

Introduction to Quantum Mechanics II

The Wavefunction

Complex numbers

Intro to Quantum Mechanics - part II

The wavefunction

Schrodinger's Wave Equation

Introduction to Quantum Mechanics II

The Schrodinger Wave Equation

The time-dependent Schrodinger equation (1932)

i\hbar\frac{\partial \Psi (x,t)}{\partial t}=-\frac{\hbar^2}{2m}\frac{\partial^2 \Psi (x,t)}{\partial x^2}+V\ \Psi(x,t)

Starring

\Psi

A complex-valued wave function, whose

nature is still under investigation to this day, but whose

norm evidently provides the correct probability density

P(x_1 \rightarrow x_2) =\int_{x_1}^{x_2} \Psi^*\ \Psi dx

&

\int_{-\infty}^{\infty} \Psi^*(x,t)\ \Psi(x,t)\ dx=1

Born's Rule

Normalization condition

Introduction to Quantum Mechanics II

The Schrodinger Wave Equation

The time-dependent Schrodinger equation (1932)

\Psi

A complex-valued wave function, whose

nature is still under investigation to this day, but whose

norm evidently provides the correct probability density

A proper must satisfy the following conditions

\Psi

Be single valued.

Be square-integrable.

Be continuous.

All first-order derivatives must be continuous.

\forall (x,t):

for a given pair of x and t, the wave function must have a unique value. This is typically satisfied for most mathematical functions anyway.

Since the norm of the wave function corresponds to the probability density, it must not be infinite over any range.

Essentially, there should be no jumps in the wave function. This is to guarantee that the momentum (which is the gradient of the wave function) is not infinite at any point.

Essentially, the wave function should be smooth. This is to guarantee that the energy is not infinite at any point.

Introduction to Quantum Mechanics II

The Schrodinger Wave Equation

The time-dependent Schrodinger equation (1932)

i\hbar\frac{\partial \Psi (x,t)}{\partial t}=-\frac{\hbar^2}{2m}\frac{\partial^2 \Psi (x,t)}{\partial x^2}+V\Psi(x,t)

Show that any wave function of the form

\Psi(x,t)=A\ e^{i(kx-\omega t)}

satisfies the TDSWE for any potential energy function V.

Introduction to Quantum Mechanics II

The Schrodinger Wave Equation

The time-dependent Schrodinger equation

\frac{\partial }{\partial t}\Psi (x,t)=(-i \omega) A\ e^{i(kx-\omega t)}

For a wave function of the form

\Psi(x,t)=A\ e^{i(kx-\omega t)}

=(-i \omega) \Psi(x,t)

=(-i 2\pi f) \Psi(x,t)

=(-i \frac{2\pi E}{h}) \Psi(x,t)

\hat{E}=i\hbar \frac{\partial}{\partial t}

i\hbar\frac{\partial }{\partial t}\Psi (x,t)=(E) \ \Psi (x,t)

\frac{\partial }{\partial x}\Psi (x,t)=(i k) A\ e^{i(kx-\omega t)}

=(i k) \Psi(x,t)

=(i \frac{2\pi}{\lambda}) \Psi(x,t)

=(i \frac{2\pi p}{h}) \Psi(x,t)

\hat{p}=-i\hbar \frac{\partial}{\partial x}

-i\hbar\frac{\partial }{\partial x}\Psi (x,t)=(p) \ \Psi (x,t)

Introduction to Quantum Mechanics II

The Schrodinger Wave Equation

The time-dependent Schrodinger equation

\hat{E}=i\hbar \frac{\partial}{\partial t}

\hat{p}=-i\hbar \frac{\partial}{\partial x}

Recall: the classical expression for the kinetic energy in terms of momentum is:

KE = \tfrac{1}{2}mv^2=\frac{1}{2m}(mv)^2=\frac{p^2}{2m}

where m is the rest mass of the particle.

The total energy of the particle is

E=\frac{p^2}{2m}+V

i\hbar\frac{\partial \Psi (x,t)}{\partial t}=-\frac{\hbar^2}{2m}\frac{\partial^2 \Psi (x,t)}{\partial x^2}+V\Psi(x,t)

E=KE + PE

i\hbar\frac{\partial }{\partial t}\Psi(x,t) = \frac{1}{2m}\left[-i\hbar\frac{\partial}{\partial x} \left(-i\hbar\frac{\partial}{\partial x} \Psi(x,t)\right)\right] + V\Psi(x,t)

Introduction to Quantum Mechanics II

The Schrodinger Wave Equation

The time-dependent Schrodinger equation

i\hbar\frac{\partial \Psi (x,t)}{\partial t}=-\frac{\hbar^2}{2m}\frac{\partial^2 \Psi (x,t)}{\partial x^2}+V\Psi(x,t)

One of the most important features of TDSWE is ...

i.e. The linear sum of solutions is also a solution!

Superposition

i\hbar\frac{\partial \Psi_1}{\partial t} = -\frac{\hbar^2}{2m}\frac{\partial^2 \Psi_1}{\partial x^2}+V\Psi_1

i\hbar\frac{\partial \Psi_2}{\partial t} = -\frac{\hbar^2}{2m}\frac{\partial^2 \Psi_2}{\partial x^2}+V\Psi_2

i\hbar\frac{\partial (\alpha \Psi_1+ \beta \Psi_2)}{\partial t} = -\frac{\hbar^2}{2m}\frac{\partial^2 (\alpha \Psi_1+ \beta \Psi_2)}{\partial x^2}+V(\alpha \Psi_1+\beta \Psi_2)

Introduction to Quantum Mechanics II

The Schrodinger Wave Equation

The time-dependent Schrodinger equation

i\hbar\frac{\partial \Psi (x,t)}{\partial t}=-\frac{\hbar^2}{2m}\frac{\partial^2 \Psi (x,t)}{\partial x^2}+V\Psi(x,t)

\Psi(x,t)=A\ e^{i(kx-\omega t)}

satisfies

since

\Psi(x,t)=\sum_j A_j\ e^{i(k_jx-\omega_jt)}

Introduction to Quantum Mechanics II

The Schrodinger Wave Equation

The time-dependent Schrodinger equation

i\hbar\frac{\partial \Psi (x,t)}{\partial t}=-\frac{\hbar^2}{2m}\frac{\partial^2 \Psi (x,t)}{\partial x^2}+V\Psi(x,t)

One of the most important features of TDSWE is ...

i.e. The linear sum of solutions is also a solution!

Superposition

i\hbar\frac{\partial \Psi_1}{\partial t} = -\frac{\hbar^2}{2m}\frac{\partial^2 \Psi_1}{\partial x^2}+V\Psi_1

i\hbar\frac{\partial \Psi_2}{\partial t} = -\frac{\hbar^2}{2m}\frac{\partial^2 \Psi_2}{\partial x^2}+V\Psi_2

i\hbar\frac{\partial (\alpha \Psi_1+ \beta \Psi_2)}{\partial t} = -\frac{\hbar^2}{2m}\frac{\partial^2 (\alpha \Psi_1+ \beta \Psi_2)}{\partial x^2}+V(\alpha \Psi_1+\beta \Psi_2)

Intro to Quantum Mechanics - part II

The wavefunction

Time-dependence

Introduction to Quantum Mechanics II

The Schrodinger Wave Equation

Separating the time and position dependence

If the potential energy function does not explicitly depend on time, then the position and time dependence of the wave function can be separated:

\Psi(x,t)=\psi(x)\ \phi(t)

i\hbar\frac{\partial \Psi (x,t)}{\partial t}=-\frac{\hbar^2}{2m}\frac{\partial^2 \Psi (x,t)}{\partial x^2}+V(x,t)\ \Psi(x,t)

\psi(x)

where is only position dependent

\phi(t)

and is only time dependent

Introduction to Quantum Mechanics II

The Schrodinger Wave Equation

Separating the time and position dependence

\Psi(x,t)=\psi(x)\ f(t)

i\hbar\frac{\partial \Psi (x,t)}{\partial t}=-\frac{\hbar^2}{2m}\frac{\partial^2 \Psi (x,t)}{\partial x^2}+V(x,t)\ \Psi(x,t)

i\hbar\frac{\partial}{\partial t} \left\{\psi(x)\ \phi(t)\right\} =-\frac{\hbar^2}{2m}\frac{\partial^2 }{\partial x^2} \left\{\psi(x)\ \phi(t)\right\}+V(x)\ \left\{\psi(x)\ \phi(t)\right\}

i\hbar\ \psi(x)\ \frac{d\ }{dt} \phi(t) =\left[-\frac{\hbar^2}{2m}\frac{d^2\ }{d x^2} \psi(x) +V(x)\ \psi(x)\right] \phi(t)

\left[-\frac{\hbar^2}{2m}\frac{d^2\ }{d x^2} \psi(x) +V(x)\ \psi(x)\right] = E\ \psi(x)

i\hbar\ \ \frac{d\ }{dt} \phi(t) = E\ \phi(t)

i\hbar\ \psi(x)\ \frac{d\ }{dt} \phi(t) =E\ \psi(x)\ \phi(t)

Introduction to Quantum Mechanics II

The Schrodinger Wave Equation

The time dependence

i\hbar\frac{d\phi(t)}{d t} = E\ \phi(t)

The time dependent part of the wavefunction satisfies the differential equation:

\phi(t)=e^{-\frac{i}{\hbar} E\ t}

whose solution is

\phi(t)=e^{- i\omega t}

for

E = \hbar \omega

Introduction to Quantum Mechanics II

The Schrodinger Wave Equation

\Psi(x,t)= e^{-i\omega t} \psi(x)

The time evolution of quantum states is in the form of a time dependent phase.

\Psi(x,t)= e^{-i\omega t} \sin(kx)

The time dependence

Intro to Quantum Mechanics - part II

The wavefunction

The time-independent Schrodinger equation

Introduction to Quantum Mechanics II

The Schrodinger Wave Equation

The time-independent Schrodinger equation

-\frac{\hbar^2}{2m}\frac{d^2 \psi (x)}{d x^2}+V(x)\ \psi(x)=E\ \psi(x)

The time-independent part of the wave function satisfies the time-independent Schrodinger wave equation

V(x)

whose solution depends on

Introduction to Quantum Mechanics II

The Schrodinger Wave Equation

The time-independent Schrodinger equation for constant potential

-\frac{\hbar^2}{2m}\frac{d^2 \psi (x)}{d x^2}=E\ \psi(x)

In a region where the potential is 0 (free particle)

\frac{d^2 \psi (x)}{d x^2}=-\frac{2m E }{\hbar^2}\ \psi(x)

Rearranging,

\psi(x) = A e^{ik x} + B e^{-ik x}

with the general solution:

\frac{d^2 \psi (x)}{d x^2}=(\pm ik)^2\ \psi(x)

Introduction to Quantum Mechanics II

The Schrodinger Wave Equation

The time-independent Schrodinger equation for constant potential

-\frac{\hbar^2}{2m}\frac{d^2 \psi (x)}{d x^2}+V_0\ \psi(x)=E\ \psi(x)

In a region where the potential is constant:

\frac{d^2 \psi (x)}{d x^2}=-\frac{2m }{\hbar^2}(E-V_0)\ \psi(x)

Rearranging,

\frac{d^2 \psi (x)}{d x^2}=+\alpha^2\ \psi(x)

case:

E \lt V_0

\psi(x) = A e^{\alpha x} + B e^{-\alpha x}

with the general solution:

\frac{d^2 \psi (x)}{d x^2}=-\alpha^2\ \psi(x)

case:

E \gt V_0

\psi(x) = A e^{i\alpha x} + B e^{-i\alpha x}

with the general solution:

Intro to Quantum Mechanics - part II

The wavefunction

Detailed Example: The infinite potential well in 1-D

Introduction to Quantum Mechanics II

The Schrodinger Wave Equation

Infinite square-well potential

Find an expression for the wave functions that satisfy the time-independent Schrodinger wave equation for a particle "trapped" in an infinite square-well potential, and their energies.

V_I = \infty

V_{III}= \infty

V_{II} = 0

Introduction to Quantum Mechanics II

The Schrodinger Wave Equation

Infinite square-well potential

Find an expression for the wave functions that satisfy the time-independent Schrodinger wave equation for a particle "trapped" in an infinite square-well potential, and their energies.

V_I = \infty

V_{III}= \infty

-\frac{\hbar^2}{2m}\frac{d^2 \psi (x)}{d x^2}+V(x)\ \psi(x)=E\ \psi(x)

\psi(x)

The wavefunction, , has to satisfy

-\frac{\hbar^2}{2m}\frac{d^2 \psi_I }{d x^2}+\infty\ \psi_I=E\ \psi_I

Region I:

x\le0

Region II:

0 \le x \le L

-\frac{\hbar^2}{2m}\frac{d^2 \psi_{II} }{d x^2}=E\ \psi_{II}

Region III:

x\ge L

-\frac{\hbar^2}{2m}\frac{d^2 \psi_{III} }{d x^2}+\infty\ \psi_{III}=E\ \psi_{III}

V_{II} = 0

Introduction to Quantum Mechanics II

The Schrodinger Wave Equation

Infinite square-well potential

Find an expression for the wave functions that satisfy the time-independent Schrodinger wave equation for a particle "trapped" in an infinite square-well potential, and their energies.

\uparrow \infty

\psi(x)

The wavefunction, , has to also be single- valued, continuous everywhere, and normalized.

@x=0:

\frac{d\psi_I}{dx}=\frac{d\psi_{II}}{dx}

\int_{-\infty}^{\infty} \psi^*(x)\ \psi(x)\ dx=1

\psi_I=\psi_{II}

\&

@x=L:

\frac{d\psi_{II}}{dx}=\frac{d\psi_{III}}{dx}

\psi_{II}=\psi_{III}

\&

@x=-\infty:

\psi_{I}=0

@x=+\infty:

\psi_{III}=0

Normalization :

Introduction to Quantum Mechanics II

The Schrodinger Wave Equation

Infinite square-well potential

Find an expression for the wave functions that satisfy the time-independent Schrodinger wave equation for a particle "trapped" in an infinite square-well potential, and their energies.

V_I = \infty

V_{III}= \infty

-\frac{\hbar^2}{2m}\frac{d^2 \psi (x)}{d x^2}+V(x)\ \psi(x)=E\ \psi(x)

-\frac{\hbar^2}{2m}\frac{d^2 \psi_I }{d x^2}+\infty\ \psi_I=E\ \psi_I

Region I:

x\le0

Region III:

x\ge L

-\frac{\hbar^2}{2m}\frac{d^2 \psi_{III} }{d x^2}+\infty\ \psi_{III}=E\ \psi_{III}

\implies \psi_I = 0

\implies \psi_{III} = 0

Introduction to Quantum Mechanics II

The Schrodinger Wave Equation

Infinite square-well potential

Find an expression for the wave functions that satisfy the time-independent Schrodinger wave equation for a particle "trapped" in an infinite square-well potential, and their energies.

V_I = \infty

V_{III}= \infty

-\frac{\hbar^2}{2m}\frac{d^2 \psi (x)}{d x^2}+V(x)\ \psi(x)=E\ \psi(x)

Region II:

0 \le x \le L

-\frac{\hbar^2}{2m}\frac{d^2 \psi }{d x^2}=E\ \psi

\implies

\frac{d^2 \psi }{d x^2}=-\frac{2mE}{\hbar^2}\psi

\implies

\frac{d^2 \psi }{d x^2}=-k^2\psi

\implies

\psi_{II}=A e^{i k x}+B e^{-i k x}

\psi_{II}=\red{A} e^{i \red{k} x}+\red{B} e^{-i \red{k} x}

-\frac{\hbar^2}{2m}\frac{d^2 \red{\psi} }{d x^2}={E}\ \red{\psi}

-\frac{\hbar^2}{2m}\frac{d^2 \red{\psi} }{d x^2}=\blue{E}\ \red{\psi}

Introduction to Quantum Mechanics II

The Schrodinger Wave Equation

Infinite square-well potential

Find an expression for the wave functions that satisfy the time-independent Schrodinger wave equation for a particle "trapped" in an infinite square-well potential, and their energies.

V_I = \infty

V_{III}= \infty

\implies

\psi_{II}=\red{A} e^{i \red{k} x}+\red{B} e^{-i \red{k} x}

@x=0:

\psi_I=\psi_{II}

@x=L:

\psi_{II}=\psi_{III}

=0

\implies\quad 0=A e^{i k \times 0}+B e^{-i k \times 0}

\implies\quad B=-A

\implies\quad 0=A e^{i k L}-A e^{-i k L}

=0

\implies\quad kL=n\pi

n=0,1,2,...

\psi_{II}=A \left( e^{i \tfrac{n\pi}{L} x}-e^{-i \tfrac{n\pi}{L} x} \right)

\implies\psi_{II}=A' \sin \left( \tfrac{n\pi}{L} x \right)

n=0,1,2,...

Introduction to Quantum Mechanics II

The Schrodinger Wave Equation

Infinite square-well potential

Find an expression for the wave functions that satisfy the time-independent Schrodinger wave equation for a particle "trapped" in an infinite square-well potential, and their energies.

V_I = \infty

V_{III}= \infty

\psi_{II}=A' \sin \left( \tfrac{n\pi}{L} x \right)

n=0,1,2,...

\int_{-\infty}^{\infty} \psi^*(x)\ \psi(x)\ dx=1

Normalization :

\int_{-\infty}^{0} \psi_I^*\ \psi_I^\ \ dx + \int_{0}^{L} \psi_{II}^*\ \psi_{II}^\ \ dx + \int_{L}^{+\infty} \psi_{III}^*\ \psi_{III}^\ \ dx =1

\int_{-\infty}^{0} 0+\int_{0}^{L} |\psi_{II}(x)|^2 dx+\int_{L}^{\infty}0=1

\int_{0}^{L} A'^2 \sin^2\left(\frac{n\pi x}{L}\right)dx=1

\psi_{II}(x)=\sqrt{\frac{2}{L}} \sin\left(\frac{n\pi x}{L}\right)

A'^2 =\frac{1}{\int_{0}^{L} \sin^2\left(\frac{n\pi x}{L}\right)dx}

Introduction to Quantum Mechanics II

The Schrodinger Wave Equation

Infinite square-well potential

Find an expression for the wave functions that satisfy the time-independent Schrodinger wave equation for a particle "trapped" in an infinite square-well potential, and their energies.

V_I = \infty

V_{III}= \infty

\psi_n(x)=\quad\quad\sqrt{\frac{2}{L}} \sin\left(\frac{n\pi }{L}x\right)

\{

0

x\le0

0 \le x\le L

x\ge L

-\frac{\hbar^2}{2m}\frac{d^2 \psi (x)}{d x^2}+V(x)\ \psi(x)=E\ \psi(x)

... satisfies TDSWE:

... is single-valued & continuous

... is square-integrable

... is normalized

@x=0:

\frac{d\psi_I}{dx}=\frac{d\psi_{II}}{dx}

\psi_I=\psi_{II}

\&

@x=L:

\frac{d\psi_{II}}{dx}=\frac{d\psi_{III}}{dx}

\psi_{II}=\psi_{III}

\&

@x=\pm\infty:

\psi_{I}(-\infty)=0

\psi_{III}(+\infty)=0

\int_{-\infty}^{\infty} \psi^*(x)\ \psi(x)\ dx=1

Introduction to Quantum Mechanics II

The Schrodinger Wave Equation

Infinite square-well potential

Find an expression for the wave functions that satisfy the time-independent Schrodinger wave equation for a particle "trapped" in an infinite square-well potential, and their energies.

V_I = \infty

V_{III}= \infty

\psi_n(x)=\quad\quad\sqrt{\frac{2}{L}} \sin\left(\frac{n\pi }{L}x\right)

\{

0

x\le0

0 \le x\le L

x\ge L

Introduction to Quantum Mechanics II

The Schrodinger Wave Equation

Infinite square-well potential

Find an expression for the wave functions that satisfy the time-independent Schrodinger wave equation for a particle "trapped" in an infinite square-well potential, and their energies.

V_I = \infty

V_{III}= \infty

-\frac{\hbar^2}{2m}\frac{d^2 \red{\psi} }{d x^2}=\blue{E}\ \red{\psi}

Recall

-\frac{\hbar^2}{2m} \left( \frac{n\pi }{L}\right)^2 \left(-\red{\psi}\right)=\blue{E}\ \red{\psi}

E_n=n^2\frac{h^2}{8mL^2}

n=1,2,3,...

\psi_n(x)=\quad\quad\sqrt{\frac{2}{L}} \sin\left(\frac{n\pi }{L}x\right)

\{

0

x\le0

0 \le x\le L

x\ge L

Introduction to Quantum Mechanics II

The Schrodinger Wave Equation

Infinite square-well potential

Find an expression for the wave functions that satisfy the time-independent Schrodinger wave equation for a particle "trapped" in an infinite square-well potential, and their energies.

V_I = \infty

V_{III}= \infty

E_n=n^2\frac{h^2}{8mL^2}

n=1,2,3,...

\psi_n(x)=\sqrt{\frac{2}{L}} \sin\left(\frac{n\pi }{L}x\right)

Intro to Quantum Mechanics - part II

The wavefunction

Example: The finite potential well in 1-D

Introduction to Quantum Mechanics II

The Schrodinger Wave Equation

finite square-well potential

Find an expression for the wave functions that satisfy the time-independent Schrodinger wave equation for a particle "trapped" in a finite square-well potential, and their energies.

V_{II} = 0

Introduction to Quantum Mechanics II

The Schrodinger Wave Equation

Finite square-well potential

Find an expression for the wave functions that satisfy the time-independent Schrodinger wave equation for a particle "trapped" in a finite square-well potential, and their energies.

-\frac{\hbar^2}{2m}\frac{d^2 \psi (x)}{d x^2}+V(x)\ \psi(x)=E\ \psi(x)

\psi(x)

The wavefunction, , has to satisfy

-\frac{\hbar^2}{2m}\frac{d^2 \psi }{d x^2}+V_0\ \psi=E\ \psi

Region I:

x\le0

Region II:

0 \le x \le L

-\frac{\hbar^2}{2m}\frac{d^2 \psi }{d x^2}=E\ \psi

Region III:

x\ge L

-\frac{\hbar^2}{2m}\frac{d^2 \psi }{d x^2}+V_0\ \psi=E\ \psi

Introduction to Quantum Mechanics II

The Schrodinger Wave Equation

Finite square-well potential

Find an expression for the wave functions that satisfy the time-independent Schrodinger wave equation for a particle "trapped" in a finite square-well potential, and their energies.

\psi(x)

The wavefunction, , has to also be single- valued, continuous everywhere, and normalized.

@x=0:

\frac{d\psi_I}{dx}=\frac{d\psi_{II}}{dx}

\int_{-\infty}^{\infty} \psi^*(x)\ \psi(x)\ dx=1

\psi_I=\psi_{II}

\&

@x=L:

\frac{d\psi_{II}}{dx}=\frac{d\psi_{III}}{dx}

\psi_{II}=\psi_{III}

\&

@x=-\infty:

\psi_{I}=0

@x=\infty:

\psi_{III}=0

Introduction to Quantum Mechanics II

The Schrodinger Wave Equation

Finite square-well potential

Find an expression for the wave functions that satisfy the time-independent Schrodinger wave equation for a particle "trapped" in a finite square-well potential, and their energies.

-\frac{\hbar^2}{2m}\frac{d^2 \psi (x)}{d x^2}+V(x)\ \psi(x)=E\ \psi(x)

-\frac{\hbar^2}{2m}\frac{d^2 \psi_{I} }{d x^2}+V_0\ \psi_{I}=E\ \psi_{I}

\frac{d^2 \psi_{I} }{d x^2}=\frac{2m(V_0-E)}{\hbar^2}\psi_{I}

\implies

\frac{d^2 \psi_{I} }{d x^2}=\alpha^2\psi_{I}

\psi_{I}=A_1\ e^{+\alpha x}+B_1\ e^{-\alpha x}

\implies

Region I:

x\le0

@x=-\infty

\psi_{I}=0

\psi_{I}=A_1\ e^{+\alpha x}

\implies

Introduction to Quantum Mechanics II

The Schrodinger Wave Equation

Finite square-well potential

Find an expression for the wave functions that satisfy the time-independent Schrodinger wave equation for a particle "trapped" in a finite square-well potential, and their energies.

-\frac{\hbar^2}{2m}\frac{d^2 \psi (x)}{d x^2}+V(x)\ \psi(x)=E\ \psi(x)

-\frac{\hbar^2}{2m}\frac{d^2 \psi_{III} }{d x^2}+V_0\ \psi_{III}=E\ \psi_{III}

\frac{d^2 \psi_{III} }{d x^2}=\frac{2m(V_0-E)}{\hbar^2}\psi_{III}

\implies

\frac{d^2 \psi_{III} }{d x^2}=\alpha^2\psi_{III}

\psi_{III}=A_3\ e^{+\alpha x}+B_3\ e^{-\alpha x}

\implies

@x=\infty

\psi_{III}=0

\psi_{III}=B_3\ e^{-\alpha x}

\implies

Region III:

x\ge L

\implies

Introduction to Quantum Mechanics II

The Schrodinger Wave Equation

Finite square-well potential

-\frac{\hbar^2}{2m}\frac{d^2 \psi (x)}{d x^2}+V(x)\ \psi(x)=E\ \psi(x)

\psi_I=A e^{+\alpha x}

Region I:

x\le0

Region III:

x\ge L

&

\psi_{III}=B e^{-\alpha x}

Find an expression for the wave functions that satisfy the time-independent Schrodinger wave equation for a particle "trapped" in a finite square-well potential, and their energies.

Introduction to Quantum Mechanics II

The Schrodinger Wave Equation

Finite square-well potential

-\frac{\hbar^2}{2m}\frac{d^2 \psi (x)}{d x^2}+V(x)\ \psi(x)=E\ \psi(x)

Region II:

0 \le x \le L

-\frac{\hbar^2}{2m}\frac{d^2 \psi }{d x^2}=E\ \psi

\implies

\frac{d^2 \psi }{d x^2}=-\frac{2mE}{\hbar^2}\psi

\implies

\frac{d^2 \psi }{d x^2}=-k^2\psi

\implies

\psi=C e^{i k x}+D e^{-i k x}

Find an expression for the wave functions that satisfy the time-independent Schrodinger wave equation for a particle "trapped" in a finite square-well potential, and their energies.

Introduction to Quantum Mechanics II

The Schrodinger Wave Equation

Finite square-well potential

-\frac{\hbar^2}{2m}\frac{d^2 \psi (x)}{d x^2}+V(x)\ \psi(x)=E\ \psi(x)

@x=0\rightarrow \psi_I=\psi_{II}

There are 5 unknowns:

A, B, C, D,

and

E

Need 5 equations:

@x=0\rightarrow \frac{d\psi_I}{dx}=\frac{d\psi_{II}}{dx}

@x=L\rightarrow \psi_{II}=\psi_{III}

@x=L\rightarrow \frac{d\psi_{II}}{dx}=\frac{d\psi_{III}}{dx}

\int_{-\infty}^{\infty} \psi^*(x)\ \psi(x)\ dx=1

Find an expression for the wave functions that satisfy the time-independent Schrodinger wave equation for a particle "trapped" in a finite square-well potential, and their energies.

Introduction to Quantum Mechanics II

The Schrodinger Wave Equation

Finite square-well potential

-\frac{\hbar^2}{2m}\frac{d^2 \psi (x)}{d x^2}+V(x)\ \psi(x)=E\ \psi(x)

Find an expression for the wave functions that satisfy the time-independent Schrodinger wave equation for a particle "trapped" in a finite square-well potential, and their energies.

Introduction to Quantum Mechanics II

The Schrodinger Wave Equation

Finite square-well potential

Find an expression for the wave functions that satisfy the time-independent Schrodinger wave equation for a particle in a finite square-well potential, and their energies.

-\frac{\hbar^2}{2m}\frac{d^2 \psi (x)}{d x^2}+V(x)\ \psi(x)=E\ \psi(x)

\delta x = \frac{\hbar}{\sqrt{ 2m (V_0-E)}}

Penetration depth

\delta x

Intro to Quantum Mechanics - part II

The wavefunction

Example: The harmonic potential well in 1-D

Introduction to Quantum Mechanics II

The Schrodinger Wave Equation

Simple harmonic oscillator

Find an expression for the wave functions that satisfy the time-independent Schrodinger wave equation for a particle experiencing a simple harmonic potential

V(x) = \tfrac{1}{2}\kappa x^2

Introduction to Quantum Mechanics II

The Schrodinger Wave Equation

Simple harmonic oscillator

Find an expression for the wave functions that satisfy the time-independent Schrodinger wave equation for a particle experiencing a simple harmonic potential

-\frac{\hbar^2}{2m}\frac{d^2 \psi (x)}{d x^2}+V(x)\ \psi(x)=E\ \psi(x)

V(x) = \tfrac{1}{2}\kappa x^2

\frac{d^2 \psi (x)}{d x^2}=-\frac{2m}{\hbar^2}\left( E-\tfrac{1}{2}\kappa x^2\right) \psi(x)

\frac{d^2 \psi }{d x^2}=\left( \alpha^2 x^2 -\beta \right) \psi

or, simply

where

\alpha^2=\frac{m\kappa}{\hbar^2}

\beta=\frac{2mE}{\hbar^2}

&

Introduction to Quantum Mechanics II

The Schrodinger Wave Equation

Simple harmonic oscillator

Find an expression for the wave functions that satisfy the time-independent Schrodinger wave equation for a particle experiencing a simple harmonic potential

V(x) = \tfrac{1}{2}\kappa x^2

\frac{d^2 \psi }{d x^2}=\left( \alpha^2 x^2 -\beta \right) \psi

Solving

with the conditions

\psi(x)\rightarrow0

as

x\rightarrow \pm \infty

\int_{-\infty}^{\infty} \psi^*(x)\ \psi(x)\ dx=1

we find

\psi_n=H_n(x)e^{-\alpha x^2}

E_n=(n+\tfrac{1}{2})\hbar\sqrt{\kappa/m}

n=0,1,2,...

Introduction to Quantum Mechanics II

The Schrodinger Wave Equation

Simple harmonic oscillator

Find an expression for the wave functions that satisfy the time-independent Schrodinger wave equation for a particle experiencing a simple harmonic potential

V(x) = \tfrac{1}{2}\kappa x^2

we find

\psi_n=H_n(x)e^{-\alpha x^2/2}

E_n=(n+\tfrac{1}{2})\hbar\sqrt{\kappa/m}

n=0,1,2,...

E_0=\tfrac{1}{2}\hbar \omega

E_1=\tfrac{3}{2}\hbar \omega

E_2=\tfrac{5}{2}\hbar \omega

Introduction to Quantum Mechanics II

The Schrodinger Wave Equation

Simple harmonic oscillator

Find an expression for the wave functions that satisfy the time-independent Schrodinger wave equation for a particle experiencing a simple harmonic potential

V(x) = \tfrac{1}{2}\kappa x^2

Intro to Quantum Mechanics - part II

The wavefunction

Expectation Values

Introduction to Quantum Mechanics II

Expectation values

Definition

For any physical observable,

\langle Q\rangle=\int_{-\infty}^{\infty} \psi^*(x)\ \hat{Q}\ \psi(x)\ dx

{Q}

, the expectation value of repeated measurements of this observable is given by

where

\hat{Q}

is a mathematical operator corresponding to the physical observable, and

\psi

is the wavefunction of the system.

Introduction to Quantum Mechanics II

Expectation values

example

Determine the expectation values

\langle x\rangle,

for a particle in the first excited state in an infinite square well.

\langle x^2\rangle,

\langle p\rangle,

\langle p^2\rangle,

\langle E\rangle

Introduction to Quantum Mechanics II

Expectation values

example

Determine the expectation values

\langle x\rangle,

for a particle in the first excited state in an infinite square well.

\langle x^2\rangle,

\langle p\rangle,

\langle p^2\rangle,

\langle E\rangle

\langle x\rangle=\int_{-\infty}^{\infty} \psi^*(x)\ \hat{x}\ \psi(x)\ dx

Introduction to Quantum Mechanics II

Expectation values

example

Determine the expectation values

\langle x\rangle,

for a particle in the first excited state in an infinite square well.

\langle x^2\rangle,

\langle p\rangle,

\langle p^2\rangle,

\langle E\rangle

Introduction to Quantum Mechanics II

Expectation values

example

\langle x^2\rangle=\int_{-\infty}^{\infty} \psi^*(x)\ \hat{x^2}\ \psi(x)\ dx

Determine the expectation values

\langle x\rangle,

for a particle in the first excited state in an infinite square well.

\langle x^2\rangle,

\langle p\rangle,

\langle p^2\rangle,

\langle E\rangle

=\int_{-\infty}^{0} +\int_{0}^{L} +\int_{L}^{\infty}

=\int_{0}^{L} \frac{2}{L} {x^2} \sin^2\left(\frac{2\pi x}{L}\right) dx

=\frac{(8\pi^2-3)L^2 }{24\pi^2}

Introduction to Quantum Mechanics II

Expectation values

example

\langle p\rangle=\int_{-\infty}^{\infty} \psi^*(x)\ \hat{p}\ \psi(x)\ dx

Determine the expectation values

\langle x\rangle,

for a particle in the first excited state in an infinite square well.

\langle x^2\rangle,

\langle p\rangle,

\langle p^2\rangle,

\langle E\rangle

Introduction to Quantum Mechanics II

Expectation values

example

\langle p^2\rangle=\int_{-\infty}^{\infty} \psi^*(x)\ \hat{p}\ \hat{p}\ \psi(x)\ dx

Determine the expectation values

\langle x\rangle,

for a particle in the first excited state in an infinite square well.

\langle x^2\rangle,

\langle p\rangle,

\langle p^2\rangle,

\langle E\rangle

\langle p^2\rangle=\frac{h^2}{L^2}

Introduction to Quantum Mechanics II

Expectation values

example

\langle E\rangle=\int_{-\infty}^{\infty} \psi^*(x)\ (\frac{\hat{p}^2}{2m}+V)\ \psi(x)\ dx

Determine the expectation values

\langle x\rangle,

for a particle in the first excited state in an infinite square well.

\langle x^2\rangle,

\langle p\rangle,

\langle p^2\rangle,

\langle E\rangle

=\int_{0}^{L} \psi^*(x)\ (\frac{\hat{p}^2}{2m}+V)\ \psi(x)\ dx

=\int_{0}^{L} \psi^*(x)\ \frac{\hat{p}^2}{2m}\ \psi(x)\ dx

+\int_{0}^{L} \psi^*(x)\ V\ \psi(x)\ dx

\langle E\rangle= \frac{\langle{p}^2\rangle}{2m}+\langle V \rangle

= \frac{1}{2m} \frac{h^2}{L^2}+0

Introduction to Quantum Mechanics II

Expectation values

example

Determine the expectation values

\langle x\rangle,

for a particle in the first excited state in an infinite square well.

\langle x^2\rangle,

\langle p\rangle,

\langle p^2\rangle,

\langle E\rangle

Alternatively,

since the energy of the first excited state in an infinite square potential well is definite (i.e. not uncertain)

repeated measurement of the energy will always yield the same outcome

then

the average of those outcomes will always come out to be the known energy of that state

thus

E_n=n^2\frac{h^2}{8mL^2}

\langle E \rangle = E_2

=\frac{h^2}{2mL^2}

=2^2\frac{h^2}{8mL^2}

Introduction to Quantum Mechanics II

Expectation values

example

Determine the expectation values

\langle x\rangle,

for a particle in the first excited state in an infinite square well.

\langle x^2\rangle,

\langle p\rangle,

\langle p^2\rangle,

\langle E\rangle

Introduction to Quantum Mechanics II

Uncertainties from Expectation values

Definition

\left(\Delta x\right)^2 = \left< x^2\right> - \left< x\right>^2

\Delta x

\left(\Delta x\right)^2

Introduction to Quantum Mechanics II

Uncertainties from Expectation values

Example

Determine the momentum uncertainty for a particle in the first excited state in an infinite square well, and use Heisenberg's uncertainty principle to bound the uncertainty in position.

Introduction to Quantum Mechanics II

Summary

The time-dependent Schrodinger equation (1932)

i\hbar\frac{\partial \Psi (x,t)}{\partial t}=-\frac{\hbar^2}{2m}\frac{\partial^2 \Psi (x,t)}{\partial x^2}+V\ \Psi(x,t)

Starring

\Psi

A complex-valued wave function, whose

nature is still under investigation to this day, but whose

norm evidently provides the correct probability density

P(x_1 \rightarrow x_2) =\int_{x_1}^{x_2} \Psi^*\ \Psi dx

&

\int_{-\infty}^{\infty} \Psi^*(x,t)\ \Psi(x,t)\ dx=1

Born's Rule

Normalization condition

Introduction to Quantum Mechanics II

Summary

Separating the time and position dependence

If the potential energy function does not explicitly depend on time, then the position and time dependence of the wave function can be separated:

\Psi(x,t)= e^{-\frac{i}{\hbar} E\ t}\ \psi(x)\

-\frac{\hbar^2}{2m}\frac{d^2 \psi (x)}{d x^2}+V(x)\ \psi(x)=E\ \psi(x)

\psi(x)

(known as stationary states) are solutions of the time-independent SWE, each with a corresponding definite energy

E

Must be single valued, and continuous everywhere. Must be square integrable. Also, first-order derivatives must be continuous everywhere.

\psi(x)

Introduction to Quantum Mechanics II

Summary

Expectation values and uncertainties

\left(\Delta Q\right)^2 = \left< Q^2\right> - \left< Q\right>^2

\langle Q\rangle=\int_{-\infty}^{\infty} \psi^*(x)\ \hat{Q}\ \psi(x)\ dx

The expectation value of any observable can be calculated for a given wave function via:

The variance of any physical quantity can be estimated from the expectation values:

Introduction to Quantum Mechanics II

Summary

Infinite square-well potential

Find an expression for the wave functions that satisfy the time-independent Schrodinger wave equation for a particle "trapped" in an infinite square-well potential, and their energies.

V_I = \infty

V_{III}= \infty

E_n=n^2\frac{h^2}{8mL^2}

n=1,2,3,...

\psi_n(x)=\sqrt{\frac{2}{L}} \sin\left(\frac{n\pi }{L}x\right)

\Psi_n(x,t)=\sqrt{\frac{2}{L}} \sin\left(\frac{n\pi }{L}x\right) \ e^{-i E_n t / \hbar}

Introduction to Quantum Mechanics II

Summary

Finite square-well potential

-\frac{\hbar^2}{2m}\frac{d^2 \psi (x)}{d x^2}+V(x)\ \psi(x)=E\ \psi(x)

Find an expression for the wave functions that satisfy the time-independent Schrodinger wave equation for a particle "trapped" in a finite square-well potential, and their energies.

Introduction to Quantum Mechanics II

Summary

Example

Suppose the wavefunction describing the position of a particle is given by a gaussian wavefunction of the form:

\int_0^{\infty}e^{-c x^2}dx=\sqrt{\frac{\pi}{4c}}

\int_0^{\infty}x^2\ e^{-cx^2}dx=\frac{1}{4c}\sqrt{\frac{\pi}{c}}

\psi(x)=C e^{-a(x+b)^2}

What is the probability of finding the particle in the range

x\in \left[b-\tfrac{1}{a}, b+\tfrac{1}{a}\right]

Does this wavefunction satisfy Heisenberg's uncertainty principle?